]*L*/2 in the localized regime; in contrast, in the presence of a mobility edge, *N*0*j* takes values between 0 and 1 at different sites.\n[density]\nA plot of *N*0 as a function of *j*/*L* and *ω**D*, shown in the left panel Fig. [density], indicates that the transition from the ergodic to the multifractal and MBL regions leaves it signature in fermion transport. We find that the steady state value of *N*0 in the MBL (high frequency) regime is ∼ ± 1 for the left and right halves of the chain respectively. This indicates that the steady state is close to the initial state as expected in the MBL regime. In contrast, *N*0 = 0 for all *j*/*L* in the ergodic (low frequency) regime which indicates that the system has reached the infinite temperature steady state as expected for a driven ergodic many-body system. In between, the system displays a range values of *N*0 for different *j* which indicates the intermediate behavior in the multifractal regime. This is also reflected in the plot of $N\\_{\\rm av}$ which shows a kink in the multifractal regime and indicates a transition between the localized and delocalized regimes. We note here that the steady state localization here happens due to both MBL nature of the Floquet eigenstates and dynamical localization due to the drive; thus fermion transport may not solely reflect MBL properties in the high frequency regime.\n### Auto-correlation function\nWe define the steady state auto correlation function at site *j* as,\n \n$$\\begin{aligned}\n\\mathcal{A}\\_s^j = ( 2 n\\_j^s -1 )( 2 n\\_j^0 -1 )\\end{aligned}$$ \nwhere,\n \n$$n\\_j^s = \\sum\\_m |c\\_{m}^{\\rm init}|^2 \\langle \\psi\\_m| \\hat\nn\\_j|\\psi\\_m\\rangle$$ \nthe steady state value of $\\langle \\hat n\\_j\\rangle$ and $n\\_j^0 =\n\\langle \\psi\\_{\\rm init}|\\hat n\\_j|\\psi\\_{\\rm init}\\rangle$ is the initial value. We average this single site operator over different sites to compute $\\mathcal{A}\\_s^{\\rm\nav}=\\frac{1}{L}\\sum\\_j\\mathcal{A}\\_s^j$.\n[steadyauto]\nThe left panel of Fig. [steadyauto] shows the plot of steady state value of the auto-correlation function as a function of *j* and *ω**D*. This value is computed by averaging over *N*0 = 50 product initial states in the basis of *H* for *L* = 10 − 14 and *N*0 = 10 such states for *L* = 16. Below ℏ*ω**D*/(*π*J0) ∼ 0.25, the value of autocorrelator remains zero indicating *n**j**s* = 1/2 for all *j* in the steady state. In contrast at high drive frequencies, one expects *n**j**s* ≃ *n**j*0 leading to ${\\mathcal A}^{\\rm av} \\simeq 1$. In between 0.25 < ℏ*ω**D*/(*π*J0) < 1, the behavior is intermediate to that of delocalized or MBL phase. Thus $0 < {\\mathcal A}\\_s < 1$ indicates that the system is in a multifractal phase. As the frequency is increased beyond that, A*s* → 1 indicating the onset of localization. As before, we point out that this localization receives contribution from both the MBL nature of the Floquet eigenstates and the dynamical localization due to the drive.\n### Number entropy\nIn this subsection we shall show the steady state behavior of *S**N* denoted by *S**N**s*. We divide the system into two subsystems *A* and *B* and we integrate over subsystem *B*. To compute the number entropy, we first denote the states in the fermion number basis as ∣*χ**k*⟩. Since these are eigenstates of the number operator, for each of them, one can compute the total number of fermions *n* in subsystem A: $n\\_{A k}= \\sum\\_{j \\in A} \\langle\n\\chi\\_k|\\hat n\\_j|\\chi\\_k\\rangle$. Using the notation of Eq. [finalstate], we can write in the steady state\n \n$$\\begin{aligned}\n|d\\_k^{s}|^2 &=& |\\langle\\chi\\_k|\\psi\\_m\\rangle|^2 = \\sum\\_m |c\\_m^{\\rm init}|^2\n|\\langle \\chi\\_k|\\psi\\_m\\rangle|^2 \\label{dsdef}\\end{aligned}$$ \nUsing Eq. [dsdef], we can then obtain\n \n$$\\begin{aligned}\np\\_{s}(n) &=& \\sum\\_{k=k^{\\prime}} |d^s\\_k|^2 \\label{psdef} \\\\\nS\\_N^{s}&=&-\\sum\\_n p\\_{s}(n) \\ln p\\_{s}(n), \\,\\, \\langle S\\_N^s\\rangle =\n\\frac{1}{N}\\sum\\_i^N S\\_N^s(i) \\nonumber\\end{aligned}$$ \nwhere *k*′ denotes all states with *n**A**k* = *n*. Here ⟨*S**N**s*⟩ denotes averaged number entropy where the average is taken over *N* product states. For an initial Neel state, we denote the entropy by S*N**s* since no averaging is involved.\n[steadySN]\nFig. [steadySN] shows the steady state behavior of *S**N* for different drive frequencies. The left panel shows the average behavior when we take ℓ product states: $|\\psi\\_{\\rm\ninit}\\rangle=|\\chi\\_{\\ell}\\rangle$. We compute ⟨*S**N**s*⟩ by averaging over these states. In the low frequency regime where ergodicity is expected, ⟨*S**N**s*⟩ is large and monotonically increases with *L*. However with increasing frequency as the system becomes non-ergodic, it decreases and there is no clear monotonicity with *L*. This is similar to the behavior seen in Ref. . A similar behavior is also seen on the right panel where $|\\psi\\_{\\rm init}\\rangle$ is taken to be the Neel state. For these plots, we have chosen the AA potential to be *V**j* = *V*0cos(2*π**η**j* + *ϕ*) and have averaged over *ϕ* to smoothen out possible local fluctuations. These local fluctuations tend to arise in *S**N* at high frequencies as the steady state values heavily depend on the local potentials near the half-chain cut. To prevent this from affecting our overall result we perform the averaging in this scenario. From the plot, we find that at large drive frequencies (*i.e*., in the localized regime), the steady state curves for different *L* almost overlap. In contrast, in the ergodic regime there is a clear increase of *S**N* with system size. The plot confirms that the system is localized at high frequencies but not at intermediate frequencies.\nFloquet Perturbation Theory\n===========================\nIn this section, we aim to obtain a semi-analytic, albeit perturbative, understanding of several features of the driven interacting AA model found via exact numerics using a Floquet perturbation theory which is known to provide accurate results at intermediate frequencies provided that the term in *H*(*t*) with largest amplitude is treated exactly. In the present case, since ${\\mathcal J}\\_0 \\gg V\\_0, V\\_{\\rm int}$, one needs to treat the drive term exactly. Thus one obtains\n \n$$\\begin{aligned}\nU\\_0(t,0) &=& e^{i {\\mathcal J}\\_0 t \\sum\\_k \\epsilon\\_k \\hat n\\_k,\n/\\hbar} \\quad t \\le T/2\n\\nonumber\\\\\n&=& e^{i{\\mathcal J}\\_0(T-t) \\sum\\_k \\epsilon\\_k \\hat n\\_k /\\hbar} \\quad\nt\\ge T/2 \\label{zeroord}\\end{aligned}$$ \nwhere *U*0 is the exact evolution operator corresponding to $H=\nH\\_0= {\\mathcal J}\\_0 \\sum\\_k \\epsilon\\_k \\hat n\\_k$ and *ε**k* = − 2cos*k* for the fermion chain with nearest-neighbor hopping. From Eq. [zeroord], we find that *U*0(*T*, 0) = *I* indicating *H**F*(0) = 0.\nNext, we compute the first order Floquet Hamiltonian *H**F*(1). To this end, we note that within first order perturbation theory, the correction to the evolution operator is given by \n \n$$\\begin{aligned}\nU\\_1(T,0) &=& \\frac{-i}{\\hbar} \\int\\_0^T dt U\\_0^{\\dagger}(t,0) (H\\_1 +\nH\\_A) U\\_0 (t,0) \\nonumber\\\\\nH\\_{F}^{(1)} &=& \\frac{i}{T} U\\_1(T,0) \\label{firstord}\\end{aligned}$$ \nThe computation of *H**F*(1) can be done in a straightforward manner following the method discussed in Refs. . The matrix elements of *H**F*(1) between Fock states in momentum space, denoted by ∣*n**k*⟩ = ∣*n**k*1(1).....*n**k**N*(*N*)⟩, is given by\n \n$$\\begin{aligned}\n{\\langlen\\_{i}|} H\\_F^{(1)}{|n\\_{f}\\rangle}&=& V\\_0 \\sum\\_{q,k\\_1} \\frac{2\\hbar\n(1-e^{ i T \\mathcal{J}\\_0 \\alpha\\_1(k\\_1,q)/\\hbar})}{T\n\\mathcal{J}\\_0 \\alpha\\_1(k\\_1,q)} f(q)\\delta\\_{n^{i}\\_{k\\_1}, n^{f}\\_{k\\_1+q}-1} \\nonumber\\\\\n&&+V\\_{\\rm int} \\sum\\_{k\\_1,k\\_2,q}\\frac{2 \\hbar(1-e^{ i T \\mathcal{J}\\_0\n\\alpha\\_2(k\\_1,k\\_2,q)/\\hbar})}{T \\mathcal{J}\\_0 \\alpha\\_2(k\\_1,k\\_2,q)}\ne^{-i q}\n\\delta\\_{n\\_{i}^{k\\_1},n\\_f^{k\\_1}-1}\\delta\\_{n\\_{i}^{k\\_2},n\\_f^{k\\_2}-1}\n\\delta\\_{n\\_{f}^{k\\_2-q},n\\_i^{k\\_2-q}+1}\\delta\\_{n\\_{f}^{k\\_1+q},n\\_i^{k\\_1+q}+1}\n\\nonumber\\\\\n\\alpha\\_1(k\\_1,q) &=& \\cos k\\_1-\\cos (k\\_1+q), \\quad\n\\alpha\\_2(k\\_1,k\\_2,q)=\\cos k\\_1+\\cos k\\_2-\\cos (k\\_1+q)-\\cos(k\\_2-q)\n\\label{FPTmom}\\end{aligned}$$ \nwhere *f*(*q*) = ∑*j*exp[ − *i**j**q*]cos(2*π**η**j*). We note that for *T* → 0, the Floquet Hamiltonian reduces to that obtained from the first order Magnus expansion $H\\_F^{(1)}(T \\to 0) \\simeq H\\_F^{\\rm\nmagnus} = H\\_A + H\\_1$. However at intermediate frequencies, the frequency dependence of *H**F*(1) is much more complicated. Moreover, a similar calculation shows that *U*2(*T*, 0) = *U*1(*T*, 0)2/2; thus *H**F*(2) = 0 and *H**F*(1) represents the sole contribution to *H**F* to ${\\rm O}(T^2)$. These features allow one to expect that it shall provide at least qualitatively accurate description of the dynamics of the system at intermediate drive frequencies.\nNext, we use Eq. [FPTmom] to numerically compute matrix elements of *H**F*(1) between Fock states in the position basis. A numerical diagonalization of the matrix thus obtained yields the eigenvalues and eigenvectors (in the real space Fock basis) for comparison with our exact results.\n[fpt]\nThe results obtained from the above-mentioned procedure is depicted in Fig. [fpt]. From Fig. [fpt] we find that the results obtained from FPT agrees with those from exact numerics discussed in Sec. [phase]. The top left panel of Fig. [fpt] shows the plot of *I**m*, obtained from eigenvectors of *H**F*(1) using Eq.[ipr1], as a function of *ω**D*. The top right panel shows the corresponding plot for *τ*2. We find that both the plots show similar multifractal behavior as seen in Figs. [fig1] and [fig2] within similar range of *ω**D*/(*π*J0). In particular, we find that the eigenvectors of *H**F*(1) exhibits delocalized eigenstates for *ω**D*/(*π*J0) ≤ 0.45, multifractal eigenstates states for 0.45 ≤ *ω**D*/(*π*J0) ≤ 1.5 and localized eigenstates for *ω**D*/(*π*J0) > 1.5. The plot of *D**q* as a function of *ω**D*, shown in the bottom left panel of Fig. [fpt], also shows qualitatively similar behavior to that obtained from ED shown in bottom right panel of Fig. [fig1]; however, we note that the change in *D**q* signifying the transition to multifractal phase is seen around $\\hbar \\omega\\_D/(\\pi {\\mathcal J}\\_0) \\simeq 0.6$. Thus the position of the transition is not accurately captured by *H**F*(1). Nevertheless, *H**F*(1) does predict the transition to the multifractal phase as seen in the bottom right panel of Fig.[fpt], where a plot of Δ*S* as a function of *ω**D* shows a distinct peak. The peak becomes sharper with increasing *L* which is consistent with the result obtained from ED. Thus we conclude that *H**F*(1), computed using FPT, constitutes a semi-analytic Floquet Hamiltonian which shows a transition from ergodic to multifractal regime at intermediate frequencies.\nDiscussion\n==========\nIn this work, we have studied a driven fermionic chain with an AA potential and nearest neighbor density-density interaction between the fermions. Our analysis constitutes a detailed study, both numerical and semi-analytic, of the Floquet Hamiltonian of such a system as a function of drive frequency in the limit of large drive amplitude.\nWe have shown that such a driven system supports multifractal many-body Floquet eigenstates for a range of drive frequencies in the intermediate drive frequency regime. We find that the eigenstates are ergodic in the low frequency and many-body localized in the high drive frequency regime. In between, for system sizes accessible in our numerics, results indicate a possible transition from ergodic to multifractal phase at $\\omega\\_D= \\omega\\_c \\simeq\n0.43 {\\mathcal J}/\\hbar$. Upon further increasing the drive frequency, the eigenstates become many-body localized via a smooth crossover. The presence of the transition from ergodic to the multifractal phase seems likely due to two reasons. First, the plot of entropy fluctuation Δ*S* as a function of *T* shows a peak at the transition which gets sharper with increasing system size. Second, *b*1 (Eq. [enteq]) changes sign at this point which is indicative of a transition from the ergodic phase. However, we need that one needs finite size numerics with larger system size to settle this issue; this has not been attempted in this work.\nOur analysis indicates that several dynamic quantities that we study can distinguish between the multifractal ergodic and MBL phases. These include the fermion auto-correlation function and the short time behavior of the normalized participation ratio. The former quantity decays sharply to zero in the ergodic phase due to spreading of the system in the Hilbert space. In the MBL phase, it remains close to its initial value since the system retains its initial memory. In the multifractal phase, we find an initial sharp decay of the auto-correlation function, followed by oscillation around a steady state value which is intermediate between its ergodic (zero) and the MBL (unity) counterparts. The latter quantity shows system-size independent behavior as a function of number of drive cycles in the ergodic phase and displays a clear *L* dependence in the MBL phase. In contrast, it shows intermediate behavior with oscillations as a function of *n*0 in the multifractal phase. We note that in contrast, the entanglement entropy and the number entropy do not distinguish between the multifractal and the MBL eigenstates.\nWe have also studied steady state properties of the driven system, starting from a domain wall initial state, by computing transport properties, auto-correlation function and the number entropy. We find all of these quantities reflect a change from localized to delocalized regime as a function of drive frequency. However, the localization seen in transport also receives contribution from dynamical localization at high drive frequencies. We also find that near the transition frequency, the distribution of the number density of fermions in the steady state acquires a large width; this suggests a possible signature of the multifractal regime in fermion transport. A similar feature is seen in the steady state value of auto-correlation function which satisfies $0 < {\\mathcal\nA}\\_s <1$ in the multifractal phase; this is in sharp contrast to its values zero and unity in the ergodic and MBL phases respectively. The plot of steady state number entropy also show a sharp drop at the transition which becomes sharper with increasing *L*.\nWe have also obtained similar qualitative features for the driven fermionic chain from a semi-analytic, albeit perturbative, Floquet Hamiltonian computed using FPT. Remarkably, this perturbative Floquet Hamiltonian reproduces multifractality of the Floquet eigenstates and also points towards a transition from the ergodic to the multifractal regime. Our results thus constitutes an analytic Floquet Hamiltonian which support ergodic, multifractal and MBL eigenstates depending on the drive frequency.\nOur results could be relevant for ultracold interacting fermions in the presence of an 1D optical lattice. The realization of the AA potential can be done using techniques discussed in Refs. and. The drive can be implemented by appropriate tuning of the strength of the laser used to create the optical lattice. We suggest measurement of density-density auto-correlation of the fermions. Our results suggest that the short time behavior of this auto-correlation function would be sufficient to distinguish between the ergodic, MBL and the multifractal phases. In particular, in the intermediate drive frequency regime, we expect the auto-correlation function to exhibit a sharp drop followed by oscillations around a finite non-zero value.\nThe fate of the multifractal phase that we obtain in the thermodynamic limit remains an open question. The phase remains stable within the system sizes that we could access within ED; however, it is possible that it might either shrink for large *L* leading to a direct ergodic-MBL quantum phase transition. An investigation of the stability of the multifractal phase in the thermodynamic limit is beyond the scope of the present paper. We note however, that the finite-sized chains that we study in this paper may possibly be experimentally realized using ultracold atoms in optical lattices.\nR.G. thanks M. Znidaric for discussions and funding from project J1-1698 Many-body transport engineering for financial support.\n99\nrev1a P. W. Anderson, Phys. Rev. **109**, 1492 (1958); E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. **42**, 673 (1979).\nrev12 P. A. Lee and T. V. Ramakrishnan, Rev. Mod. Phys. **57**, 287 (1985).\naaref S. Aubry and G. Andre, Ann. Isr. Phys. Soc. **3**, 133 (1980).\naaref2 M. Ya. Azbel, Sov. Phys. JETP **17**, 665 (1963).\naaref3 M. Ya. Abzel, *ibid.* **19**, 634 (1964).\naaref4 M. Ya Abzel, Phys. Rev. Lett. **43**, 1954 (1979).\ngaarefJ. Biddle, B. Wang, J. D. J. Priour, and S. D. Sarma, Phys. Rev. A, **80**, 021603(R) (2009).\ngaaref2 D. J. Boers, B. Goedeke, D. Hinrichs, and M. Holthaus, Phys. Rev. A, **75**, 063404 (2007).\ngaaref3R. Riklund, Y. Liu, G. Wahlstrom, and Z. Zhao-bo, J. Phys. C: Solid State Phys., **19**, L705 (1986).\ngaaref4F. A. B. F. de Moura, A. V. Malyshev, M. L. Lyra, V. A. Malyshev, and F. Dominguez-Adame, Phys. Rev. B, **71**, 174203 (2005).\ngaaref5A. V. Malyshev, V. A. Malyshev, and F. Dominguez- Adame, Phys. Rev. B, **70**, 172202 (2004).\ngaaref6S.-J. Xiong and G.-P. Zhang, Phys. Rev. B, **68**, 174201 (2003).\ngaaref7A. Rodriguez, V. A. Malyshev, G. Sierra, M. A. Martin-Delgado, J. Rodríguez-Laguna, and F. Domínguez-Adame, Phys. Rev. Lett. **90**, 027404 (2003).\ngaaref8S. Das Sarma, A. Kobayashi, and R. E. Prange, Phys. Rev. Lett. **56**, 1280 (1986).\ngaaref9J. Biddle and S. Das Sarma, Phys. Rev. Lett. **104**, 070601 (2010).\nsinharef X. Deng, S. Ray, S. Sinha, G. V. Shlyapnikov, and L. Santos,, 025301 (2019).\notherref D-L Deng, S. Ganeshan, X. Li, R. Modak, S. Mukerjee, and J. H. Pixley, Ann. Phys. **529**, 1600399 (2017).\nZu Z. Xu, H. Huangfu, Y. Zhang and S.Chen, New J. Phys. **22**, 013036 (2020).\notherref2 A. Jagannathan, arXiv:2012.14744 (unpublished).\nexp1 G. Roati, C. D. Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, Nature (London) **453**, 895 (2008).\nexp12 B. Deissler, M. Zaccanti, G. Roati, C. D. Errico, M. Fattori, M. Modugno, G. Modugno, and M. Inguscio, Nat. Phys. **6**, 87 (2010).\nexp2 M. Schreiber, S. S. Hodgman, P. Bordia, H. P. L¨schen, M. H. Fischer, R. Vosk, E. Altman, U. Schneider, and I. Bloch, Science **349**, 842 (2015).\nexp22 H. P. Lauschen, P. Bordia, S. Scherg, F. Alet, E. Altman, U. Schneider, and I. Bloch, Phys. Rev. Lett. **119**, 260401 (2017).\nrev1J. Dziarmaga, Adv. Phys. **59**, 1063 (2010).\nrev2 A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, Rev. Mod. Phys. **83**, 863 (2011).\nA. Dutta, G. Aeppli, B. K. Chakrabarti, U. Divakaran, T. F. Rosenbaum, and D. Sen, *Quantum phase transitions in transverse field spin models: from statistical physics to quantum information* (Cambridge University Press, Cambridge, 2015).\nrev4 S. Mondal, D. Sen, and K. Sengupta, *Quantum Quenching, Annealing and Computation*, edited by Das, A., Chandra, A. & Chakrabarti, B. K. Lecture Notes in Physics, Vol. 802 (Springer, Berlin, Heidelberg, 2010), Chap. 2, p. 21; C. De Grandi and A. Polkovnikov, *ibid*, Chap 6, p. 75.\nrev5 M. Bukov, L. D’Alessio and A. Polkovnikov, Adv. Phys. **64**, 139 (2015).\nL. D’Alessio and A. Polkovnikov, Ann. Phys. **333**, 19 (2013).\nL. D’Alessio, Y. Kafri, A. Polokovnikov, and M. Rigol, Adv. Phys. **65**, 239 (2016).\nrev8I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. **80**, 885 (2008); L. Taurell and L. Sanchez-Palencia, C. R. Physique **19**, 365 (2018).\nrev9 A. Sen, D. Sen, and K. Sengupta, arXiv:2102.00793 (unpublished).\ndynloc1a T. Nag, S. Roy, A. Dutta, and D. Sen, Phys. Rev. B **89**, 165425 (2014).\ndynloc1b T. Nag, D. Sen, and A. Dutta, Phys. Rev. A **91**, 063607 (2015).\ndynloc1c A. Agarwala, U. Bhattacharya, A. Dutta, and D. Sen, Phys. Rev. B **93**, 174301 (2016).\ndynloc1d A. Agarwala and D. Sen, Phys. Rev. B **95**, 014305 (2017).\ndynloc2a D. J. Luitz, Y. Bar Lev, and A. Lazarides, SciPost Phys. **3**, 029 (2017).\ndynloc2bD. J. Luitz, A. Lazarides, and Y. Bar Lev, Phys. Rev. B **97**, 020303 (2018)\ndynloc3 R. Ghosh, B. Mukherjee, and K. Sengupta, Phys. Rev. B **102**, 235114(2020).\ndynf1aA. Das, Phys. Rev. B **82**, 172402 (2010).\ndynf1b S Bhattacharyya, A Das, and S Dasgupta, Phys. Rev. B **86** 054410 (2010).\ndynf1c S. Hegde, H. Katiyar, T. S. Mahesh, and A. Das, Phys. Rev. B **90**, 174407 (2014)\ndynf2aS. Mondal, D. Pekker, and K. Sengupta, Europhys. Lett. **100**, 60007 (2012).\ndynf2bU. Divakaran and K. Sengupta, Phys. Rev. B **90**, 184303 (2014); B. Mukherjee.\ndynf3 B. Mukherjee, A. Sen, D. Sen, and K. Sengupta, Phys. Rev B **102**, 075123 (2020).\neth1J. M. Deutsch, Phys. Rev. A **43**, 2046 (1991).\neth2 M. Srednicki, Phys. Rev. E **50**, 888 (1994); *ibid.*, J. Phys. A **32**,1163 (1999).\neth3 M. Rigol, V. Dunjko, and M. Olshanii, Nature (London) **452**, 854 (2008).\neth4 L. D’Alessio and M. Rigol, M. Phys. Rev. X **4**, 041048 (2014).\nethv0 C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papic, Z. Nat. Phys. 14, 745 (2018).\nethv01 C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papic, Phys. Rev. B **98**,155134 (2018).\nethv02 K. Bull, I. Martin, and Z. Papic, Phys. Rev. Lett. **123**, 030601 (2019).\nethv03 V. Khemani, C. R. Lauman, and A. Chandran, Phys. Rev. B **99**, 161101 (2019).\nethv04 S. Maudgalya, N. Regnault, and B. A. Bernevig, Phys. Rev. B **98**, 235156 (2018).\nethv05 T. Iadecola, M. Schecter, and S. Xu, arXiv:1903.10517 (unpublished).\nethv06 N. Shiraishi, arXiv:1904.05182 (unpublished).\nethv07 M. Schecter, and T. Iadecola, arXiv:1906.10131 (unpublished).\nethv08 P. A. McClarty, M. Haque, A. Sen, and J. Richter, Phys. Rev. B **102**, 224303 (2020).\nethv09 D. Banerjee and A. Sen, Phys. Rev. Lett. **126**, 220601 (2021).\nethv1 B. Mukherjee, S. Nandy, A. Sen, D. Sen and K. Sengupta, Phys. Rev B **101**, 245107 (2020).\nethv2B. Mukherjee, A. Sen, D. Sen and K. Sengupta, Phys. Rev B **102**, 014301 (2020).\nmbl1 D. M. Basko, I. L. Aleiner, B. L. Altshuler, Ann. Phys. **321**, 1126-1205 (2006).\nmbl11 V. Oganesyan and D. A. Huse, Phys. Rev. B **75**, 155111 (2007).\nmbl12 A. Pal and D. A. Huse, Phys. Rev. B **82**, 174411 (2010).\nmbl13 D. A. Huse, R. Nandkishore, V. Oganesyan, A. Pal, and S. L. Sondhi, Phys. Rev. B **88**, 014206 (2013).\nmbl14 R.Vosk and E. Altman, Phys. Rev. Lett. **110**, 067204 (2013).\nmbl15 M.Serbyn, Z. Papic, and D. A. Abanin, Phys. Rev. Lett. **111**, 127201 (2013).\nmbl16 D. A. Huse, R. Nandkishore, and V. Oganesyan, Phys. Rev. B **90**, 174202 (2014).\nmbl17 T. Grover, arXiv:1405.1471 (unpublished).\nmbl18 M.Serbyn, Z. Papic, and D. A. Abanin, Phys. Rev. X **5**, 041047 (2015).\nmbl19 K. Agarwal, S. Gopalakrishnan, M. Knap, M. Mueller, and E. Demler, Phys. Rev. Lett. **114**, 160401 (2015).\nmbl20 V. Khemani, S. P. Lim, D. N. Sheng, and D. A. Huse, Phys. Rev. X **7**, 021013 (2017).\nmbl21 J. H. Bardarson, F. Pollmann, J. E. Moore, Phys. Rev. Lett. **109**, 017202 (2012).\nmbl22 S. Iyer, V. Oganesyan, G. Refael, and D. A. Huse, Phys. Rev. B **87**, 134202 (2013).\nmbl23 J. A. Kjall, J. H. Bardarson, and F. Pollmann, Phys. Rev. Lett. **113**, 107204 (2014).\nmbl24 R.Vasseur, S. A. Parameswaran, and J. E. Moore, Phys. Rev. B **91**, 140202 (2015).\nmbl25 M. Serbyn, Z. Papic, and D. A. Abanin, Phys. Rev. B **90**, 174302 (2014).\nmbl26 D. Pekker, G. Refael, E. Altman, E. Demler, and V. Oganesyan, Phys. Rev. X **4**, 011052 (2014).\nmbl27 R. Modak and S. Mukerjee, Phys. Rev. Lett. **115**, 230401 (2015).\nmbl28 D. J. Luitz, N. Laflorencie, and F. Alet, Phys. Rev. B **91**, 081103 (2015).\nmbl29 W. De Roeck, F. Huveneers, M. Mueller, and M. Schiulaz, Phys. Rev. B **93**, 014203 (2016).\nmbl30 R. Modak, S. Ghosh and S. Mukerjee, Phys. Rev. B **97**, 104204 (2018).\nmbl31 M. Schreiber, S. S. Hodgman, P. Bordia, H. P. Luschen, M. H. Fischer, R. Vosk, E. Altman, U. Schneider, and I. Bloch, Science **349**, 842 (2015).\nrafael S. Iyer, G. Refael, V. Oganesyan, and D. A. Huse, Phys. Rev. B, **87**, 134202 (2013).\nabanin P. Ponte, Z. Papic, F. Huveneers, and D. A. Abanin, Phys. Rev. Lett. **114**, 140401 (2015).\npal A. Pal and D. A. Huse, Phys. Rev. B, **82**, 174411 (2010).\nhuse V. Oganesyan and D. A. Huse, Phys. Rev. B, **75**, 155111 (2007).\nznidaric1 M. Znidaric, T. Prosen, and P. Prelovsek, Phys. Rev. B, **77**, 064426 (2008).\nrigol L. D’Alessio and M. Rigol, Phys. Rev. X, **4**, 041048 (2014).\nchalker S. J. Garratt and J. T. Chalker, arXiv:2012.11580.\nlin J. Lindinger, A. Buchleitner, and A. Rodriguez, Phys. Rev. Lett. **122**, 106603 (2019).\ngopalakrishnan H. Singh, B. Ware, R. Vasseur, S. Gopalakrishnan, arXiv:2101.04126 (unpublished).\nmukherjee A. Dutta, S. Mukherjee, and K. Sengupta,, 144205 (2018).\nneqref1 S. Roy, I. M. Khyamovich, A. Das, and R. Moessner, Scipost Phys. **4**, 025 (2018).\nneqref2 S. Ray, A. Ghosh, and S. Sinha Phys. Rev. E **97**, 010101(R) (2018).\nsarkar M. Sarkar, R. Ghosh, A. Sen, and K. Sengupta, Phys. Rev. B, **103**, 184309 (2021).\ncritphase Y. Wang, C. Cheng, X-J Liu, and D. Yu, Phys. Rev. Lett. **126**, 080602 (2021).\nalet N. Mace, F. Alet, and N. Laorencie, Phys. Rev. Lett. **123**, 180601 (2019).\nluitz D. J. Luitz, I. M. Khaymovich, and Y. Bar Lev, SciPost Phys. Core **2**, 006 (2020).\ngarg Y. Prasad and A. Garg, Phys. Rev. B. **103**, 064203 (2021).\nabanin2 I. V. Protopopov, R. K. Panda, T. Parolini, A. Scardicchio, E. Demler, and D. A. Abanin, Phys. Rev. X **10**, 011025 (2020).\nsantos1A. Soló rzano, L. F. Santos, and E. J. Torres-Herrera, arXiv:2102.02824 (unpublished).\nsid S. Gopalakrishnan and S. A. Parameswaran, Phys. Rep. **862**, 1, (2020).\nnag R. Modak and T. Nag Phys. Rev. Research **2**, 012074 (2020).\nbloch P. Bordia, H. Lüschen, U. Schneider, M. Knap and I. Bloch, Nat. Phys. **13**, 460 (2017).\nBera T. L. M. Lezama, S. Bera, and J. H. Bardarson, Phys. Rev, B **99**, 161106 (2019).\ndassarma X. Li and S. Das Sarma, Phys. Rev. B **101**, 064203(2020).\nswingle Y. Yoo, J. Lee, and B. Swingle Phys. Rev. B **102**, 195142 (2020).\ndas S. Roy, I. M. Khyamovich, A. Das, and R. Moessner, Scipost Phys. **4**, 025 (2018).\nkrylov A. Nauts and R. E. Wyatt, Phys. Rev. Lett. **51**, 2238 (1983).\nmfracentr G. De Tomasi and I. M. Khaymovich, Phys. Rev. Lett. **124**, 200602(2020).\nroop R. Ghosh and A. Das, Phys. Rev. B **103**, 024202 (2021).\nnumberentrp M. Kiefer-Emmanouilidis, R. Unanyan, M. Fleischhauer, and J. Sirker, Phys. Rev. Lett. **124**, 243601 (2020).\nnumberentrA. Lukin, M. Rispoli, R. Schittko, M. E. Tai, A. M. Kaufman, S. Choi, V. Khemani, J. Leonard, and M. Greiner, Science **364**, 256 (2019).\ninhomquench T. Rakovszky, C. W. von Keyserlingk, and F. Pollmann, Phys. Rev. B **100**, 125139 (2019).\nnumberentrp2 M. Kiefer-Emmanouilidis, R. Unanyan, M. Fleischhauer, and J. Sirker, Phys. Rev. B **103**, 024203 (2021).\n"},"data_id":{"kind":"string","value":"arxiv_0185554"}}},{"rowIdx":46394,"cells":{"content":{"kind":"string","value":"Parity-violating scalar-tensor theory and the Qi-Xiu\n====================================================\nIntroduction\n============\nWhile the dark sector of current cosmology remains insufficiently understood, new physics beyond general relativity (GR) and the standard model of particle physics, including the strong, electromagnetic, and weak interactions, is required. Various observable effects, such as helicity asymmetry and beta decay, have been observed, indicating the presence of parity violation in weak interactions. In the era of precision cosmology, the polarization features of the cosmic microwave background reveal clues about parity violation beyond the standard model. Consequently, when turning our attention to gravity, it is natural to question if there are any parity-violating effects in gravitational interactions.\nFrom a theoretical perspective, incorporating parity-violating terms opens up new avenues for understanding fundamental aspects of gravity. At the quantum level, parity-violating terms would contribute to the renormalization and ultraviolet behavior of the theory, thus potentially providing insights into the quantization of gravity or the final unified theory. From an observational perspective, the detection of gravitational wave events, together with forthcoming gravitational wave experiments, opens a new window and provides an opportunity to directly test these parity-violating theories and models. Different polarization modes in a cosmological background would be exhibited in these parity-violating theories, and modifications to the dispersion relations would also be introduced. Moreover, as corrections to standard inflationary models, parity-violating terms change the polarization modes of the resulting primordial gravitational waves, which should produce non-vanishing TB/EB correlations in the cosmic microwave background polarization. The resulting behavior of GWs in different parity-violating gravity models has been extensively studied.\nFor the purpose of constructing gravitational theories with parity violation, the parity-violating terms can enter the gravitational action through various approaches. A certain class of quintessence or axion fields would generate such a parity-violating feature by coupling the pseudo-scalar with electromagnetism. Another notable example is the four-dimensional Chern-Simons modified gravity, in which the Pontryagin term with a coefficient depending on a scalar field is added to the standard Einstein-Hilbert action. Generalization of Chern-Simons gravity in the more general scalar-tensor theories is considered in, in which various couplings between the Riemann tensor and covariant derivatives of the scalar field up to the second order are introduced. A totally antisymmetric Levi-Civita tensor (volume form) *ɛ**μ**ν**ρ**σ* appears in the action and is used to construct various parity-violating terms. One lesson from the construction of these parity-violating terms in the framework of scalar-tensor theory is that the resulting theories are ghost-free only if the scalar field possesses a timelike gradient so that the so-called unitary gauge can be taken.\nOn the other hand, generally covariant scalar-tensor (GST) theories in the unitary gauge take the form of metric theories with Lorentz violation and, more specifically, spatially covariant gravity (SCG) theories respecting only spatial covariance. The SCG theories have simpler forms and are, in fact, more convenient in the cosmological context. Well-studied examples, although originally motivated by different purposes, include the Hořava-Lifshitz gravity, as well as the effective field theory of inflation/dark energy. The parity-violating effects in Hořava gravity have been investigated in. A general class of SCG theories was proposed in as an alternative and more unifying approach to introducing a scalar degree of freedom and generalizing the scalar-tensor theories. It was further extended by introducing a dynamical lapse function, nonmetricity, as well as an auxiliary scalar field. Constraints from cosmological perturbations and gravitational waves on SCG have been explored in.\nThere are several advantages to working in the framework of SCG theories. Firstly, in the case of a single scalar degree of freedom, there is a one-to-one correspondence between theories of GST and SCG through gauge-fixing/gauge-recovering procedures. Based on the classification of GST monomials in, such a correspondence has been discussed in detail in, and the explicit linear mappings between GST and SCG monomials are developed in (see also ). In other words, SCG should not be viewed merely as Lorentz-violating theories but as a general and unifying description of scalar-tensor theories. Secondly, thanks to its dependence only on spatial diffeomorphism, the separation of temporal and spatial derivatives allows us to generally include the parity-violating terms with odd-order spatial derivatives and linear terms of higher-order time derivatives in the action without introducing ghost DoFs. This is also reflected in the fact that the absence of ghosts in SCG (i.e., in the unitary gauge) is a necessary condition for the corresponding GST theory to be ghost-free (i.e., in any gauge). The specific behavior of theories that are degenerate in the unitary gauge is discussed in. Therefore, SCG provides a unifying framework to study gravity theories with parity violation in a systematic manner.\nIn this work, we aim to investigate parity-violating scalar-tensor theories without ghosts in the unitary gauge. We concentrate on the GST Lagrangians of the polynomial type. Instead of starting from the most general GST polynomial and studying its behavior in the unitary gauge so that all the ghost-like terms get canceled, we will make use of the correspondence between GST and SCG. That is, we will construct the most general SCG monomials with parity violation and then find their generally covariant correspondence. The parity-violating SCG monomials have been studied in without Lie derivatives of the lapse function and the extrinsic curvature. However, these terms naturally arise in GST terms in the unitary gauge. Therefore, a complete analysis of both GST and SCG monomials up to *d* = 4 with *d* being the number of derivatives in the unitary gauge is needed. Then, by making use of the covariant correspondence, we can get the desired parity-violating scalar-tensor theories without ghosts in the unitary gauge. This work is thus devoted to these issues.\nThe paper is organized as follows. In Sec. [sec:scg], we briefly review the general framework of SCG theory. We also show a specific SCG model in Sec. [subsec:ndlapse] with parity-violating terms of the polynomial type as an illustration of the construction and classification of the SCG monomials. In Sec. [sec:pvscg], we list the basic building blocks for SCG by introducing new operators involving Lie derivatives of the lapse function and the extrinsic curvature. We then exhaust all the possible parity-violating SCG monomials up to *d* = 4 with *d* being the total number of derivatives. In Sec. [sec:gfgst], we connect the ghost-free SCG model with parity-violating scalar-tensor theories. In particular, we get a set of 7 parity-violating scalar-tensor Lagrangians that are ghost-free in the unitary gauge. Finally, Sec. [sec:con] is devoted to conclusions.\nThroughout this paper, we use *i*, *j*, *k*, ⋯ to denote spatial indices in a coordinate basis, and *a*, *b*, *c*, ⋯ to denote spacetime indices in a general basis. Curvature tensors such as *R**a**b**c**d* and *R**a**b* denote the 4-dimensional quantities, while 3 *R**a**b* and 3 *R**i**j* etc. denote the 3-dimensional (spatial) quantities.\nSpatially covariant gravity\n===========================\nGeneral framework\n-----------------\nThe Lagrangian of SCG is built of metric variables and is only invariant under 3-dimensional spatial diffeomorphisms, which breaks time diffeomorphism. The action of such theories can be constructed in terms of scalars under spatial diffeomorphism. Consequently, it is natural and convenient to employ the Arnowitt-Deser-Misner (ADM) variables, which are based on the foliation structure of spacetime.\nThe general action for SCG, which extends the original proposal in by allowing higher-order temporal derivatives, takes the form\n \n$$S=\\int \\mathrm{d}t\\mathrm{d}^{3}x\\,N\\sqrt{h}\\mathcal{L}\\left(t,N,h\\_{ij},{}^{3}\\!R\\_{ij},\\nabla\\_{i},\\pounds\\_{\\bm{u}},\\varepsilon\\_{ijk}\\right), \\label{SCG\\_ori}$$ \nwhere *N* is the lapse function, *h**i**j* is the 3-dimensional spatial metric, *R**i**j* is the 3-dimensional spatial Ricci tensor, ∇*i* is the covariant derivative compatible with *h**i**j*, $\\bm{u}$ is the normal vector of the spatial hypersurfaces[1](#fn1). In order to introduce the parity-violating terms, the Levi-Civita tensor $\\varepsilon\\_{ijk}\\equiv\\sqrt{h}\\epsilon\\_{ijk}$ with *ε*123 = 1 is included. Please note the shift vector *N**i* should not be included explicitly, since it is not a genuine geometrical quantity characterizing the foliation structure but merely encodes the gauge degrees of freedom of spatial diffeomorphism.\nThe formulation in terms of the ADM variables and the spacetime decomposition makes the kinetic terms be introduced in a natural and transparent manner. In particular, the separation of the temporal and spatial derivatives in the SCG allows us to focus on higher-order time derivatives, which could possibly introduce unwanted ghost-like or unwanted degrees of freedom without any restrictions. As mentioned above, the basic variables in SCG are the lapse function *N* and the spatial metric *h**i**j*. In the original proposal of, only the extrinsic curvature *K**i**j*, which is the kinetic term for *h**i**j*, is included in the Lagrangian. However, since time diffeomorphism is broken in SCG, *N* should not be treated as an auxiliary field anymore. In fact, *N* and *h**i**j* are independent and should be treated on equal footing, and thus the temporal derivatives of both variables should enter the Lagrangian through the Lie derivatives $\\pounds\\_{\\bm{u}}$, even up to higher orders.\nFollowing, the velocity of the lapse function *N*, namely $\\dot{N}$, enters in the Lagrangian through\n \n$$F\\equiv \\frac{1}{N}\\pounds\\_{\\bm{u}}N=\\frac{1}{N^{2}}\\left(\\dot{N}-N^{i}\\nabla\\_{i}N\\right)\\,, \\label{Fdef}$$ \nwith a dot denoting the time derivative ∂*t*. The velocity of the spatial metric *h**i**j*, namely $\\dot{h}\\_{ij}$, enters the Lagrangian through the extrinsic curvature *K**i**j* defined by[2](#fn2)\n \n$$K\\_{ij}=\\frac{1}{2}\\pounds\\_{\\bm{u}}h\\_{ij}=\\frac{1}{2N}\\left(\\dot{h}\\_{ij}-\\nabla\\_{j}N\\_{i}-\\nabla\\_{i}N\\_{j}\\right)\\,.$$ \nThe resulting action is given by\n \n$$S=\\int \\mathrm{d}t\\mathrm{d}^{3}x\\,N\\sqrt{h}\\mathcal{L}\\left(t,N,h\\_{ij},F,K\\_{ij},{}^{3}\\!R\\_{ij},\\nabla\\_{i},\\varepsilon\\_{ijk}\\right)\\:. \\label{SCG1}$$ \nIn the action, *N* and *h**i**j* both act as dynamical variables, with no higher-order time derivatives appearing in the action. Other derivatives are purely spatial, which automatically evade the unwanted ghostlike mode. Generally speaking, there are four dynamical degrees of freedom (DoFs), consisting of two tensor and two scalar DoFs. Extensive research has been conducted on the degeneracy conditions under which the number of dynamical DoFs reduces to three (and to two, i.e., without any scalar DoF)[3](#fn3). In other words, additional conditions must be applied to ensure that only a single scalar DoF is present. It has been demonstrated that, up to the quadratic order in *K**i**j* and *F*, the resulting theory can be reduced to the one without *F* via disformal transformations.\nIn principle, higher-order temporal derivatives, such as $\\pounds\\_{\\bm{u}}^2 N$ and $\\pounds\\_{\\bm{u}}^2 h\\_{ij}$ can also be considered, although generally they will introduce unwanted or ghost-like DoFs. Nevertheless, in this work, we will consider the Lie derivative of *N* in terms of *F* defined in ([Fdef]) and the second-order Lie derivative of *h**i**j* in terms of $\\pounds\\_{\\bm{u}} K\\_{ij}$, which will be discussed in detail in Sec. [sec:pvscg].\nSCG with non-dynamical lapse function\n-------------------------------------\nThe SCG theory proposed in (with parity-violating extension) is described by the action\n \n$$S=\\int \\mathrm{d}t\\mathrm{d}^{3}x\\,N\\sqrt{h}\\mathcal{L}\\left(t,N,h\\_{ij},K\\_{ij},{}^{3}\\!R\\_{ij},\\nabla\\_{i},\\varepsilon\\_{ijk}\\right),\n \\label{SCG2}$$ \nin which the only time derivative entering in the Lagrangian is encoded in *K**i**j*. The theory has been proved to possess 3 DoFs through a Hamiltonian analysis [4](#fn4).\nFor the purpose of obtaining concrete 3-DoF ghost-free models, SCG Lagrangians of the polynomial type are considered and investigated in. In order to get general theoretical forms while keeping the number of monomials finite, additional restrictions are necessary to exhaust all possible monomials. In, it is assumed that the total number of derivatives does not exceed 4. As we will see later, this implies that the Riemann curvature tensor is up to the quadratic order, the derivatives (both spatial and Lie derivatives) of *N* and *h**i**j* are up to second order. Degeneracy conditions reducing 3-DoF SCG theories to 2-DoF SCG theories can be found in by different approaches. It has been demonstrated that, under specific conditions, this type of SCG can exhibit only two tensorial DoFs without propagating any scalar mode.\nIt is necessary to make a classification of the possible SCG monomials due to their large number. We will follow the same conventions and classification as in. By the Stueckelberg trick, an SCG term can be mapped to a corresponding generally covariant scalar-tensor (GST) term. Accordingly, we can assign each SCG term the set of integers (*c*0; *d*2, *d*3) to characterize the GST combination, where *c*0 is the number of Riemann curvature tensors, *d*2, *d*3 are the numbers of the second and the third covariant derivatives of *ϕ*, respectively[5](#fn5). Schematically, we can write\n \n$$\\begin{aligned}\n K\\_{ij} & \\sim a\\_{i}\\sim(0;1,0),\\\\\n {}^{3}\\!R\\_{ij} & \\sim(1;0,0),\\\\\n \\nabla\\_{k}K\\_{ij} & \\sim\\nabla\\_{j}a\\_{i}\\sim(0;0,1).\\end{aligned}$$ \nThus, the total number of derivatives in each monomial is given by\n \n$$\\begin{aligned}\n d=\\sum\\_{n=0}\\left[(n+2)c\\_{n}+(n+1)d\\_{n+2}\\right].\\end{aligned}$$ \nSince the resulting GST terms would arise only from the combinations of GST monomials of the same values of *d*, the number *d* can be used as a generic label characterizing monomials of both GST and SCG, which makes the correspondences more transparent. In this work, we consider monomials with *d* ≤ 4. Even considering higher-order operators, one would find that no new independent monomials are present in each category, as long as the value of the total number of derivatives *d* is not larger than 4.\nBased on ([SCG2]), a specific ghost-free SCG Lagrangian of the polynomial type with parity violation was proposed in. The parity-violating terms are denoted by $\\tilde{\\mathcal{L}}^{(3)}$ for *d* = 3 and $\\tilde{\\mathcal{L}}^{(4)}$ for *d* = 4, which are given by\n \n$$\\mathcal{\\tilde{L}}^{(3)}= c\\_{1}^{(0;1,1)}\\varepsilon\\_{ijk}K\\_{l}^{i}\\nabla^{j}K^{kl},$$ \nand\n \n$$\\begin{aligned}\n\\mathcal{\\tilde{L}}^{(4)} & = & c\\_{1}^{(0;2,1)}\\varepsilon\\_{ijk}K^{im}K^{jn}\\nabla\\_{m}K\\_{n}^{k}+c\\_{2}^{(0;2,1)}\\varepsilon\\_{ijk}K^{mn}K\\_{m}^{i}\\nabla^{j}K\\_{n}^{k} \\nonumber \\\\ \n & & +c\\_{3}^{(0;2,1)}\\varepsilon\\_{ijk}K\\_{l}^{i}a^{j}\\nabla^{k}a^{l}+c\\_{4}^{(0;2,1)}\\varepsilon\\_{ijk}K\\_{l}^{i}\\nabla^{j}K^{kl}K \\nonumber \\\\\n& & +c\\_{1}^{(1;2,0)}\\varepsilon\\_{ijk}{}^{3}\\!R\\_{l}^{i}K^{jl}a^{k}+c\\_{1}^{(1;0,1)}\\varepsilon\\_{ijk}{}^{3}\\!R\\_{l}^{i}\\nabla^{j}K^{kl},\\end{aligned}$$ \nrespectively. The coefficients *c**m*(*c*0; *d*2, *d*3) are generally functions of *t* and *N*.\nIn the next section, we will extend the above Lagrangian by introducing new parity-violating terms involving Lie derivatives of the lapse function and the extrinsic curvature.\nParity-violating monomials in SCG\n=================================\nIn this section, by considering novel SCG operators involving higher-order derivatives, we will construct and classify the parity-violating SCG monomials in a systematic way under the restriction *d* ≤ 4, with *d* being the total number of derivatives.\nBuilding blocks\n---------------\nFirst of all, we need to extend the set of operators for our purpose. Fundamental geometrical quantities, including the lapse function *N*, the spatial metric *h**i**j*, the acceleration *a**i*, the extrinsic curvature *K**i**j*, and the spatial Ricci tensor 3 *R**i**j*, along with their spatial derivatives, are involved as the basic ingredients. It should be noted that these quantities are associated with the foliation defined by the scalar field once the time coordinate is fixed. Moreover, we will introduce higher-order derivatives of the lapse function *N* and the spatial metric *h**i**j*. Precisely, spatial and Lie derivatives of *K**i**j*, *a**i*, and *F*, i.e., ∇*k**K**i**j*, ∇*i**a**j*, ∇*i**F*, $\\pounds\\_{\\bm{u}}K\\_{ij}$, and $\\pounds\\_{\\bm{u}}F$, as well as $\\pounds\\_{\\bm{u}}a\\_{i}$, will be taken into account in our construction. The reason for introducing these higher-order derivative operators is not only because they would extend the SCG construction, but also because they naturally (or necessarily) arise in the unitary gauge for GST monomials up to *d* = 4.\nWhen evaluating Lie derivatives with respect to the normal vector $\\bm{u}$, it is necessary to transform the expressions into a generally covariant form. Once it is established that these operators are indeed spatial tensors, we can then take their spatial components with indices *i*, *j*, ⋯. For example, the Lie derivative of the extrinsic curvature with lower indices is a spatial tensor since $n^{b}\\pounds\\_{\\bm{u}}K\\_{ab}=n^{b}\\pounds\\_{\\bm{u}}K\\_{ba}=0$. More explicitly, we have\n \n$$\\pounds\\_{\\bm{u}}K\\_{ij}=\\frac{1}{N}\\left[\\dot{K}\\_{ij}-\\left(N^{k}\\nabla\\_{k}K\\_{ij}+K\\_{kj}\\nabla\\_{i}N^{k}+K\\_{ki}\\nabla\\_{j}N^{k}\\right)\\right],$$ \nwhich involves the second order time derivative of *h**i**j* through $\\dot{K}\\_{ij}$. On the other hand, one can show that Lie derivative of the acceleration can be reduced,\n \n$$\\pounds\\_{\\bm{u}}a\\_{i}=\\pounds\\_{\\bm{u}}\\left(\\frac{\\nabla\\_{i}N}{N}\\right)=\\nabla\\_{i}F+Fa\\_{i},$$ \nwhich implies that $\\pounds\\_{\\bm{u}}a\\_{i}$ is not an independent operator. A subtle point arises when taking Lie derivatives of spatial tensors with upper indices[6](#fn6). To avoid confusion, we only use the Lie derivative of spatial tensors with lower indices in SCG. Lie derivatives of spatial tensors with mixed or upper indices are understood as merely shorthands. For example, only the Lie derivative $\\pounds\\_{\\bm{u}}K\\_{ij}$ (spatial component of $\\pounds\\_{\\bm{u}}K\\_{ab}$) is defined in SCG, while $\\pounds\\_{\\bm{u}}K\\_{j}^{i}$ is just a shorthand that stands for\n \n$$\\pounds\\_{\\bm{u}}K\\_{j}^{i}\\equiv h^{ik}\\pounds\\_{\\bm{u}}K\\_{kj}=h^{ik}e\\_{\\phantom{a}k}^{c}e\\_{\\phantom{b}j}^{b}\\pounds\\_{\\bm{u}}K\\_{cb}=e\\_{a}^{\\phantom{a}i}e\\_{\\phantom{b}j}^{b}h^{ac}\\pounds\\_{\\bm{u}}K\\_{cb}.$$ \nSimilarly, $\\pounds\\_{\\bm{u}}K^{ij}\\equiv h^{ik}h^{jl}\\pounds\\_{\\bm{u}}K\\_{kl}$, etc. Note we also have a useful relation for the spatial derivative of the acceleration ∇*i**a**j*:\n \n∇*i**a**j* = ∇*i*∇*j*ln*N* = ∇*j*∇*i*ln*N* = ∇*j**a**i*, \nwhich implies that its indices are symmetric.\nMoreover, up to *d* = 4, SCG monomials involving third-order or fourth-order derivatives can be reduced to equivalent combinations of the same order of *d* but with lower-order operators through integration by parts. This allows for the construction of a simplified and complete set of basis monomials, facilitating analysis and computations.\nFinally, we have a complete set of operators for our purpose. The lapse function *N* and the induced metric *h**i**j* can be regarded as the 0-th order operator. The independent operators up to the second order in derivatives are shown in Table [SCG operators].\n[h]\n|c|ccc|ccc| 0 order & *h**i**j* & & & *N* & & \n1st order & *K**i**j* & & & *F*, & & *a**k* \n2nd order & $\\quad \\quad \\pounds\\_{\\bm{u}}K\\_{ij}$, & ∇*k**K**i**j*, & 3 *R**i**j* & $\\quad \\quad \\pounds\\_{\\bm{u}}F,$ & ∇*k**F*, & ∇*i**a**j* \n… & … & & & … & & \n[SCG operators]\nAs we have mentioned above, up to *d* = 4 we do not need to consider operators involving derivatives higher than the second order (acting on the ADM variables), such as $\\pounds\\_{\\bm{u}}{}^{3}\\!R\\_{ij}$, $\\pounds\\_{\\bm{u}}^{2}K\\_{ij}$, ∇*i*∇*j**K**i**j*, etc.\nComplete basis for the parity-violating monomials\n-------------------------------------------------\nTo identify all the parity-violating terms constructed by these building blocks, we divide the entire process into two steps. The first step involves determining all the possible types of combinations of the given order. In this step, we do not distinguish various contractions from each other. For notational simplicity, we use the initial letters to represent each geometric quantity, such as *a* for *a**k*, *K* for *K**i**j*, *R* for 3 *R**i**j*, $\\pounds K$ for $\\pounds\\_{\\bm{u}}K\\_{ij}$, ∇*K* for ∇*k**K**i**j*, etc. In the second step, we systematically explore all the possible contractions and identify all the independent monomials. Note that the spatial Levi-Civita tensor *ɛ**i**j**k* is necessary in order to construct monomials with parity violation.\n### *d* = 3\nThere are no parity-violating monomials in the cases of *d* = 0, 1, 2. So we start from the case of *d* = 3. In the case of *d* = 3, we have two types of operator combinations, schematically denoted by [1 + 2] and [1 + 1 + 1]. Here the integers denote orders of derivatives in each building block (operator). For example, a monomial built from contracting *F* and ∇*i**F* will be of [1 + 2] type, since *F* contains the first-order derivative and ∇*i**F* contains the second-order derivative of *N*, respectively.\nFor the type of [1 + 2], besides the combinations [*K*∇*K*], [*a*∇*a*], and [*a**R*] that have been considered in, we have the following new combinations\n \n$$[F\\nabla F],\\quad [a\\pounds F],\\quad [K\\nabla F], \\quad [F\\nabla K],\\quad [a\\pounds K],$$ \ndue to the presence of *F* and $\\pounds\\_{\\bm{u}}K\\_{ij}$. Due to the (anti)symmetry of indices, no viable monomials arise from these parity-violating combinations except [*K*∇*K*]. Therefore, we refer to these cases as being “empty” and can disregard them in the subsequent step. The other type of operator combination, i.e., [1 + 1 + 1], contains the combination [*a**a**a*] as well as the new ones[7](#fn7)\n \n[*a**K**F*], [*a**F**F*]. \nClearly, no viable parity-violating monomials emerge from these combinations.\nAs a result, it becomes evident that the only non-empty parity-violating combination is [*K*∇*K*]. Moreover, the only independent contraction within this combination is:\n \n[*K*∇*K*] : *ɛ**i**j**k**K**l**i*∇*j**K**k**l*, \nwhich has already been studied in. In brief, in the case of *d* = 3, despite the introduction of new operators *F* and $\\pounds\\_{\\bm{u}}K\\_{ij}$, no new parity-violating SCG monomial emerges.\n### *d* = 4\nIn the case of *d* = 4, we have three different types of operator combinations: [2 + 2], [1 + 1 + 2], and [1 + 1 + 1 + 1]. Based on the same notation, we express all the possible combinations of each type.\nFor the type of [1 + 1 + 1 + 1], besides the combinations [*a**K**K**K*], [*a**a**a**K*], we have the following new combinations due to the presence of *F*,\n \n[*K**K**a**F*], [*a**a**a**F*], [*K**a**F**F*], [*a**F**F**F*] . \nNo viable parity-violating monomials exist in these types of combinations.\nFor the type of [1 + 1 + 2], there are viable parity-violating monomials from the combination types\n \n[*a**K*∇*a*], [*a**a*∇*K*], [*K**K*∇*K*], [*a**K**R*], \nwhich have been considered in. In addition, there are new viable parity-violating monomials corresponding to the combinations\n \n$$[aK \\pounds K],\\quad [FK\\nabla K].$$ \nWhile the following combinations\n \n$$\\begin{aligned}\n&[Ka\\pounds F],\\quad [Fa\\pounds K],\\quad [Fa\\pounds F],\\quad [Fa\\nabla a],\\quad [FaR],\\\\\n &[FF\\nabla K],\\quad [KK\\nabla F],\\quad [KF\\nabla F],\\quad [FF\\nabla F],\\quad [aa\\nabla F]\\:,\\end{aligned}$$ \nyield no viable monomials.\nFor the type of [2 + 2], besides the combinations [*R*∇*K*] and [∇*a*∇*K*], we have one new combination\n \n$$[\\pounds K\\nabla K],$$ \nwith empty cases \n$$[\\pounds F\\nabla F],\\quad [\\pounds K\\nabla F],\\quad [\\pounds F\\nabla K],\\quad [R\\nabla F],\\quad [\\nabla a\\nabla F]\\:.$$ \nIn summary, due to the presence of *F* and $\\pounds\\_{\\bm{u}}K\\_{ij}$, there arise 3 new parity-violating SCG monomials in the case of *d* = 4, which are\n \n$$\\begin{aligned}\n[FK\\nabla K] & : &\\quad \\{\\varepsilon\\_{ijk}FK\\_{l}^{i}\\nabla^{j}K^{kl}\\},\\\\{}\n[aK\\pounds K] & : &\\quad \\{\\varepsilon\\_{ijk}a^{i}K^{jl}\\pounds\\_{\\bm{u}}K\\_{l}^{k}\\},\\\\{}\n[\\pounds K\\nabla K] & : &\\quad \\{\\varepsilon\\_{ijk}\\pounds\\_{\\bm{u}}K^{li}\\nabla^{j}K\\_{l}^{k}\\}.\\end{aligned}$$ \nAt this point, we note that not all these terms are independent. One can check that the last term, i.e., $\\varepsilon\\_{ijk}\\pounds\\_{\\bm{u}}K^{li}\\nabla^{j}K\\_{l}^{k}$, is not independent and can be expressed by linear combinations of the other terms up to total derivatives. The details can be found in Appendix [app:rel]. Therefore, in the following, we do not need to consider $\\varepsilon\\_{ijk}\\pounds\\_{\\bm{u}}K^{li}\\nabla^{j}K\\_{l}^{k}$.\n[h]\n[SCGpvmono]\nWe list all the irreducible and reducible parity-violating SCG monomials in Table [SCGpvmono]. In the case of *d* = 3, there is only one parity-violating monomial. In the case of *d* = 4, there are 8 independent parity-violating monomials. Following the terminology developed in, the complete basis of parity-violating SCG polynomials for *d* = 3 is thus\n \nZ3 = {*ɛ**i**j**k**K**l**i*∇*j**K**k**l*}, \nwhich is composed of a single unfactorizable and irreducible monomial. In other words, we have dim(Z3) = 1. The “enlarged” basis for the parity-violating SCG polynomials for *d* = 4 is given by\n \n$$\\begin{aligned}\n \\mathcal{Z}\\_{4} & = & \\Big\\{\\varepsilon\\_{ijk}{}^{3}\\!R\\_{l}^{i}K^{jl}a^{k},\\varepsilon\\_{ijk}K^{im}K^{jn}\\nabla\\_{m}K\\_{n}^{k},\\varepsilon\\_{ijk}K^{mn}K\\_{m}^{i}\\nabla^{j}K\\_{n}^{k},\\varepsilon\\_{ijk}K\\_{l}^{i}a^{j}\\nabla^{k}a^{l},\\nonumber \\\\\n & & \\varepsilon\\_{ijk}{}^{3}\\!R\\_{l}^{i}\\nabla^{j}K^{kl},\\varepsilon\\_{ijk}KK\\_{l}^{i}\\nabla^{j}K^{kl},\\varepsilon\\_{ijk}a^{i}K^{jl}\\pounds\\_{\\bm{u}}K\\_{l}^{k},\\varepsilon\\_{ijk}FK\\_{l}^{i}\\nabla^{j}K^{kl}\\Big\\}, \\label{Z4}\\end{aligned}$$ \nwith dim(Z4) = 8. Note that the monomial in ([Z3]), as well as the first six monomials in ([Z4]), contain no *F* or $\\pounds\\_{\\bm{u}}K\\_{ij}$ and thus are ghost-free. However, the last two monomials in ([Z4]) involve *F* or $\\pounds\\_{\\bm{u}}K\\_{ij}$ (although in a linear manner) and thus possibly suffer from the Ostrogradsky ghost even in the SCG.\nGhost-free scalar-tensor theories with parity violation\n=======================================================\nThe GST monomials are systematically classified, and the complete basis for GST polynomials up to *d* = 4 is derived in. We refer to Appendix [app:GSTmono] for the explicit expressions of the parity-violating GST monomials, which we denote *F**n*(*c*0; *d*2, *d*3) for short. Our purpose is to find the ghost-free combinations of these GST monomials. To this end, we will first derive the corresponding expressions of these parity-violating monomials in the unitary gauge, which explicitly show the dangerous terms $\\sim \\pounds\\_{\\bm{u}}K\\_{ij}$ and *F*, thus giving us guidance on how to build the ghost-free combinations.\nBefore proceeding, we would like to clarify the notation in this work. For our purpose, we frequently switch among different but equivalent formulations of the theory. First, we refer to SCG, in which all the quantities are spatial tensors with spatial indices *i*, *j*, *k*, ⋯. For example, the actions ([SCGori]) and ([SCG1]) are of the SCG form. Sometimes it is convenient or even necessary to work with the generally covariant correspondence of the SCG terms. For example, for the extrinsic curvature, we may work with *K**a**b* instead of *K**i**j*, where the former is understood as $K\\_{ab} = \\frac{1}{2} \\pounds\\_{\\bm{u}} h\\_{ab}$ with *u**a* = − *N*∇*a**ϕ* and *h**a**b* = *g**a**b* + *u**a**u**b*. In principle, it is essentially a generally covariant scalar-tensor theory term, but “wrapped” in terms of hypersurface geometrical quantities. Following the strategy in, we refer to expressions in such a form as the “unitary gauge” or “u.g.” form for short[8](#fn8).\nDecomposition in the unitary gauge\n----------------------------------\nFirst of all, there are 4 parity-violating GST monomials that do not contain higher-order time derivatives in the unitary gauge, i.e., without the dangerous terms $\\pounds\\_{\\bm{u}}K\\_{ab}$ or *F*, which are\n \n$$F\\_{1}^{(1;1,0)}\\xlongequal{\\text{u.g.}}2\\varepsilon\\_{abc}K^{ad}\\mathrm{D}^{b}K\\_{d}^{c}, \\label{F11011dec}$$ \n \n$$F\\_{6}^{(0;2,1)}\\xlongequal{\\text{u.g.}}-\\varepsilon\\_{abc}K\\_{d}^{a}a^{b}\\mathrm{D}^{c}a^{d},$$ \n \n$$F\\_{2}^{(1;2,0)}\\xlongequal{\\mathrm{u.g.}}\\varepsilon\\_{abc}\\left(-2a^{a}a^{d}\\mathrm{D}^{c}K\\_{d}^{b}+2K\\_{d}^{a}K^{de}\\mathrm{D}^{c}K\\_{e}^{b}\\right),\\label{F1202dec}$$ \nand\n \n$$F\\_{7}^{(1;2,0)}\\xlongequal{\\mathrm{u.g.}}2\\varepsilon\\_{abc}a^{a}a^{d}\\mathrm{D}^{c}K\\_{d}^{b},\\label{F1207dec}$$ \nwith $\\sigma\\equiv \\sqrt{-\\phi\\_{a}\\phi^{a}}$. Here and in the following, we define\n \n*ɛ**a**b**c* ≡ *u**d**ɛ**d**a**b**c*, \nand “u.g.” denotes equality in the unitary gauge. In deriving the above expressions, no integration by parts has been performed. As a result, we can conclude that these 4 GST monomials are automatically free of the Ostrogradsky ghost in the unitary gauge.\nBefore proceeding, note that the SCG monomials in the decomposition of *F*2(1; 2, 0) and *F*7(1, 2, 0) in ([F1202dec]) and ([F1207dec]) are not precisely the SCG monomials chosen in the complete basis ([Z4]). Nevertheless, they can be recast as linear combinations of the SCG monomials in the complete basis up to total derivatives. The relevant expressions are given in ([F1202ibp]) and ([F1207ibp]). We also refer to Appendix [app:ibp] for more detailed integrations by parts we have used in deriving the above and in the following expressions. At this point, note that the expressions of *F*6(0; 2, 1) and *F*7(1; 2, 0) in the unitary gauge differ by a total derivative term. Indeed, this also happens for their general covariant expressions ([app:F0216red]), which implies that although *F*6(0; 2, 1) is algebraically independent, it is not independent in the sense of integration by parts.\nAccording to Appendix [app:GSTmono], there are in total 10 parity-violating GST monomials in the (1; 2, 0) category. Besides *F*2(1; 2, 0) and *F*7(1; 2, 0) mentioned above, the remaining 8 monomials in the (1; 2, 0) category will contribute to terms linear in *F* and/or $\\pounds\\_{\\bm{u}}K\\_{ij}$ in the unitary gauge. The following three monomials would contribute to the [*F**K*∇*K*] term:\n \n$$F\\_{4}^{(1;2,0)}\\xlongequal{\\mathrm{u.g.}}\\varepsilon\\_{abc}\\left(2a^{a}K^{db}\\,^{3}\\!R\\_{d}^{c}-2a^{a}a^{d}\\mathrm{D}^{c}K\\_{d}^{b}-2F\\mathrm{D}^{a}K\\_{d}^{b}K^{cd}\\right), \\label{F1204dec}$$ \n \n$$F\\_{9}^{(1;2,0)} \\xlongequal{\\text{u.g.}} \\varepsilon\\_{abc}\\left(2FK^{ad}\\mathrm{D}^{b}K\\_{d}^{c}-2KK^{ea}\\mathrm{D}^{b}K\\_{e}^{c}\\right),$$ \nand\n \n$$F\\_{10}^{(1;2,0)} \\xlongequal{\\text{u.g.}} -2\\varepsilon\\_{abc}FK^{ad}\\mathrm{D}^{b}K\\_{d}^{c}.$$ \nSCG monomials in the decomposition of *F*4(1; 2, 0) in ([F1204dec]) have not been in the form of the complete basis ([Z4]). Similar to *F*2(1; 2, 0) and *F*7(1, 2, 0) above, it can be recast as a linear combination of monomials in the complete basis up to a total derivative, which is given in ([F1204ibp]). The following three monomials contribute to the $[aK\\pounds K]$ term:\n \n$$F\\_{5}^{(1;2,0)} \\xlongequal{\\text{u.g.}} \\varepsilon\\_{abc}\\left(-K^{ad}K^{be}\\mathrm{D}\\_{e}K\\_{d}^{c}-K\\_{d}^{a}a^{b}\\mathrm{D}^{c}a^{d}-a^{a}K^{db}h^{ec}\\pounds\\_{\\boldsymbol{u}}K\\_{de}\\right),$$ \n \n$$F\\_{6}^{(1;2,0)} \\xlongequal{\\text{u.g.}} \\varepsilon\\_{abc}\\left(-\\,{}^{3}\\!R\\_{d}^{a}K^{db}a^{c}+K\\_{d}^{a}a^{b}\\mathrm{D}^{c}a^{d}+a^{a}K^{db}h^{ec}\\pounds\\_{\\boldsymbol{u}}K\\_{de}\\right),$$ \nand\n \n$$F\\_{8}^{(1;2,0)} \\xlongequal{\\text{u.g.}} \\varepsilon\\_{abc}\\left(-K\\_{d}^{a}a^{b}\\mathrm{D}^{c}a^{d}-a^{a}K^{db}h^{ec}\\pounds\\_{\\boldsymbol{u}}K\\_{de}\\right).$$ \nFinally, the following two monomials would contribute to both [*F**K*∇*K*] and $[aK\\pounds K]$ terms:\n \n$$\\begin{aligned}\n F\\_{1}^{(1;2,0)} & \\xlongequal{\\mathrm{u.g.}} & 4\\varepsilon\\_{abc}\\Big(a^{a}K^{db}\\,^{3}\\!R\\_{d}^{c}-a^{a}K\\_{d}^{c}\\mathrm{D}^{b}a^{d}-a^{a}a^{d}\\mathrm{D}^{c}K\\_{d}^{b}-F\\mathrm{D}^{a}K\\_{d}^{b}K^{cd}\\nonumber \\\\\n & & \\qquad-K^{ad}K^{be}\\mathrm{D}\\_{e}K\\_{d}^{c}-a^{a}K^{db}h^{ec}\\pounds\\_{\\boldsymbol{u}}K\\_{de}\\Big), \\label{F1201dec}\n \\end{aligned}$$ \n \n$$\\begin{aligned}\n F\\_{3}^{(1;2,0)} & \\xlongequal{\\mathrm{u.g.}} & 2\\varepsilon\\_{abc}\\left(a^{a}K\\_{d}^{c}\\mathrm{D}^{b}a^{d}+a^{a}a^{d}\\mathrm{D}^{c}K\\_{d}^{b}+F\\mathrm{D}^{a}K\\_{d}^{b}K^{cd}+a^{a}K^{db}h^{ec}\\pounds\\_{\\boldsymbol{u}}K\\_{de}\\right).\\qquad \\label{F1203dec}\n \\end{aligned}$$ \nAgain, ([F1201dec]) and ([F1203dec]) can be recast as linear combinations of monomials in the complete basis through integrations by parts, which are given in ([F1201ibp]) and ([F1203ibp]), respectively. Thus, within the (1; 2, 0) category, in order to make the theory healthy in the unitary gauge, we obtain three special combinations that cancel both dangerous terms $[aK\\pounds K]$ and [*F**K*∇*K*], such as *F*1(1; 2, 0) − 2*F*4(1; 2, 0) + 4*F*6(1; 2, 0), *F*3(1; 2, 0) + *F*4(1; 2, 0) − 2*F*6(1; 2, 0) and *F*5(1; 2, 0) + *F*6(1; 2, 0). These combinations are totally healthy in the unitary gauge with any coefficient functions of *ϕ* and *X*.\nThere are in total 5 monomials of the (2; 0, 0) category. It is easy to show that all the monomials in this category would produce dangerous terms of the $[\\pounds K\\nabla K]$ form. We have\n \n$$\\begin{aligned}\n F\\_{1}^{(2;0,0)} & \\xlongequal{\\mathrm{u.g.}} & 16\\varepsilon\\_{abc}\\Big(\\mathrm{D}^{a}a^{d}\\mathrm{D}^{c}K\\_{d}^{b}+a^{a}a^{d}\\mathrm{D}^{c}K\\_{d}^{b}+\\,{}^{3}\\!R^{da}\\mathrm{D}^{c}K\\_{d}^{b}+K\\_{d}^{a}K^{de}\\mathrm{D}^{c}K\\_{e}^{b}\\nonumber \\\\\n & & \\qquad-K^{da}K^{eb}\\mathrm{D}\\_{e}K\\_{d}^{c}-h^{da}\\mathrm{D}^{c}K^{eb}\\pounds\\_{\\boldsymbol{u}}K\\_{de}\\Big), \\label{F2001dec}\n \\end{aligned}$$ \n \n$$F\\_{2}^{(2;0,0)}\\xlongequal{\\mathrm{u.g.}}8\\varepsilon\\_{abc}\\Big(\\mathrm{D}^{a}a^{d}\\mathrm{D}^{c}K\\_{d}^{b}+a^{a}a^{d}\\mathrm{D}^{c}K\\_{d}^{b}+K\\_{d}^{a}K^{de}\\mathrm{D}^{c}K\\_{e}^{b}-h^{da}\\mathrm{D}^{c}K^{eb}\\pounds\\_{\\boldsymbol{u}}K\\_{de}\\Big),$$ \n \n$$\\begin{aligned}\n F\\_{3}^{(2;0,0)} & \\xlongequal{\\mathrm{u.g.}} & 4\\varepsilon\\_{abc}\\Big(-\\mathrm{D}^{a}a^{d}D^{c}K\\_{d}^{b}-{}^{3}\\!R^{da}\\mathrm{D}^{c}K\\_{d}^{b}-a^{a}a^{d}\\mathrm{D}^{c}K\\_{d}^{b}\\nonumber \\\\\n & & \\qquad-K\\_{d}{}^{a}K^{de}\\mathrm{D}^{c}K\\_{e}^{b}+K^{da}K^{eb}\\mathrm{D}\\_{e}K\\_{d}{}^{c}+h^{da}\\mathrm{D}^{c}K^{eb}\\pounds\\_{\\boldsymbol{u}}K\\_{de}\\Big),\n \\end{aligned}$$ \n \n$$\\begin{aligned}\n F\\_{4}^{(2;0,0)} & \\xlongequal{\\mathrm{u.g.}} & 2\\varepsilon\\_{abc}\\Big(-\\mathrm{D}^{a}a^{d}\\mathrm{D}^{c}K\\_{d}^{b}+{}^{3}\\!R^{da}\\mathrm{D}^{c}K\\_{d}^{b}-a^{a}a^{d}\\mathrm{D}^{c}K\\_{d}^{b}\\nonumber \\\\\n & & \\qquad-2K\\_{d}^{a}K^{de}\\mathrm{D}^{c}K\\_{e}^{b}+KK^{ea}\\mathrm{D}^{c}K\\_{e}^{b}+h^{da}\\mathrm{D}^{c}K^{eb}\\pounds\\_{\\boldsymbol{u}}K\\_{de}\\Big),\n \\end{aligned}$$ \n \n$$F\\_{5}^{(2;0,0)}\\xlongequal{\\mathrm{u.g.}}2\\varepsilon\\_{abc}\\Big(\\mathrm{D}^{a}a^{d}\\mathrm{D}^{c}K\\_{d}^{b}+a^{a}a^{d}\\mathrm{D}^{c}K\\_{d}^{b}+K\\_{d}^{a}K^{de}\\mathrm{D}^{c}K\\_{e}^{b}-h^{da}\\mathrm{D}^{c}K^{eb}\\pounds\\_{\\boldsymbol{u}}K\\_{de}\\Big). \\label{F2005dec}$$ \nAgain, ([F2001dec])-([F2005dec]) can be recast as linear combinations of monomials in the complete basis through integrations by parts, which are given in ([F2001ibp])-([F2005ibp]), respectively. Thus, within the (2; 0, 0) category, it is easy to find some special combinations without the $[\\pounds K\\nabla K]$ term, such as $F\\_{2}^{(2;0,0)}-\\frac{1}{2}F\\_{1}^{(2;0,0)}$,$F\\_{3}^{(2;0,0)}+\\frac{1}{4}F\\_{1}^{(2;0,0)}$,$F\\_{4}^{(2;0,0)}+\\frac{1}{8}F\\_{1}^{(2;0,0)}$ and $F\\_{5}^{(2;0,0)}-\\frac{1}{8}F\\_{1}^{(2;0,0)}$.\nGhost-free scalar-tensor polynomials\n------------------------------------\nBased on the covariant scalar-tensor terms corresponding to the SCG terms developed in, the covariant correspondence of the SCG monomials, including the parity-violating monomials, has been shown in. However, in, only the matrices of the linear mapping are shown, instead of the explicit expressions for the covariant correspondence. Moreover, the higher-order Lie derivatives have not been taken into account in. In the following, we will show the explicit expressions for the covariant correspondence of the parity-violating SCG monomials up to *d* = 4.\nFor *d* = 3, the only parity-violating SCG monomial is given in ([Z3]), of which the covariant correspondence is \n \n$$\\begin{aligned}\n \\varepsilon\\_{ijk}K\\_{l}^{i}\\nabla^{j}K^{kl} & \\rightarrow & \\varepsilon^{abcd}u\\_{a}K\\_{b}^{e}\\mathrm{D}\\_{c}K\\_{de}\\nonumber \\\\\n & = & \\frac{1}{2\\left(2X\\right)^{3/2}}\\varepsilon^{abcd}R\\_{cdef}\\phi\\_{a}\\phi^{e}\\phi\\_{b}^{f}\\equiv\\frac{1}{2}F\\_{1}^{(1;1,0)}. \\label{d3cor}\\end{aligned}$$ \n([d3cor]) can be compared with ([F11011dec]).\nFor *d* = 4, the first 6 monomials in the complete basis ([Z4]) do not involve *F* or $\\pounds\\_{u}K\\_{ij}$, the covariant correspondence of which is\n \n$$\\begin{aligned}\n \\varepsilon\\_{ijk}{}^{3}\\!R\\_{l}^{i}K^{jl}a^{k} & \\rightarrow & \\varepsilon^{abcd}u\\_{a}a\\_{b}K\\_{c}^{e}\\,{}^{3}\\!R\\_{ed}\\nonumber \\\\\n & = & -\\frac{1}{4X^{2}}\\varepsilon^{abcd}R\\_{af}\\phi\\_{b}\\phi^{e}\\phi\\_{ed}\\phi\\_{fc}-\\frac{1}{8X^{3}}\\varepsilon^{abcd}R\\_{cmef}\\phi\\_{a}\\phi^{n}\\phi^{m}\\phi^{e}\\phi\\_{d}^{f}\\phi\\_{bn}\\nonumber \\\\\n & \\equiv & -F\\_{6}^{(1;2,0)}-F\\_{8}^{(1;2,0)}, \\label{d4cor1}\\end{aligned}$$ \n \n$$\\begin{aligned}\n \\varepsilon\\_{ijk}K^{im}K^{jn}\\nabla\\_{m}K\\_{n}^{k} & \\rightarrow & \\varepsilon^{abcd}u\\_{a}K\\_{b}^{e}K\\_{c}^{f}\\mathrm{D}\\_{f}K\\_{de}\\nonumber \\\\\n & = & -\\frac{1}{8X^{3}}\\varepsilon^{abcd}\\phi\\_{a}\\phi^{e}\\left(-2XR\\_{efcm}\\phi\\_{b}^{f}\\phi\\_{d}^{m}+R\\_{efcm}\\phi^{m}\\phi^{n}\\phi\\_{nb}\\phi\\_{d}^{f}\\right)\\nonumber \\\\\n & \\equiv & F\\_{5}^{(1;2,0)}-F\\_{8}^{(1;2,0)},\\end{aligned}$$ \n \n$$\\begin{aligned}\n \\varepsilon\\_{ijk}K^{mn}K\\_{m}^{i}\\nabla^{j}K\\_{n}^{k} & \\rightarrow & \\varepsilon^{abcd}u\\_{a}K\\_{b}^{f}K\\_{f}^{e}\\mathrm{D}\\_{c}K\\_{de}\\nonumber \\\\\n & = & -\\frac{1}{16X^{3}}\\varepsilon^{abcd}\\phi\\_{a}\\phi^{e}\\left(2XR\\_{efcd}\\phi\\_{b}^{m}\\phi\\_{m}^{f}+R\\_{cdef}\\phi^{n}\\phi^{m}\\phi\\_{m}^{f}\\phi\\_{nb}\\right)\\nonumber \\\\\n & \\equiv & -\\frac{1}{2}\\left(F\\_{2}^{(1;2,0)}+F\\_{7}^{(1;2,0)}\\right),\\end{aligned}$$ \n \n$$\\begin{aligned}\n \\varepsilon\\_{ijk}K\\_{l}^{i}a^{j}\\nabla^{k}a^{l} & \\rightarrow & \\varepsilon^{abcd}u\\_{a}K\\_{r}^{b}a^{c}\\mathrm{D}^{d}a^{r}\\nonumber \\\\\n & = & -\\frac{1}{8X^{3}}\\varepsilon^{abcd}\\phi\\_{a}\\phi^{e}\\phi^{f}\\phi\\_{eb}\\phi\\_{c}^{r}\\phi\\_{drf}\\nonumber \\\\\n & \\equiv & -F\\_{6}^{(0;2,1)}=\\frac{1}{2}F\\_{7}^{(1;2,0)}-\\nabla\\_{d}\\left(\\frac{1}{\\sigma^{6}}\\varepsilon\\_{abc}^{\\phantom{abc}d}\\phi^{e}\\phi^{f}\\phi^{a}\\phi\\_{e}^{b}\\phi^{cm}\\phi\\_{mf}\\right),\\end{aligned}$$ \nwhere we have used ([app:F0216red]),\n \n$$\\begin{aligned}\n \\varepsilon\\_{ijk}R\\_{l}^{i}\\nabla^{j}K^{kl} & \\rightarrow & \\varepsilon^{abcd}u\\_{a}\\,{}^{3}\\!R\\_{b}^{e}\\mathrm{D}\\_{c}K\\_{de}\\nonumber \\\\\n & = & \\frac{1}{16X^{3}}\\varepsilon^{abcd}\\Big\\{-4X^{2}R\\_{a}^{e}R\\_{cdef}\\phi^{f}\\phi\\_{b}-2X\\phi\\_{b}\\phi^{m}\\phi^{f}\\phi^{n}R\\_{am\\phantom{e}n}^{\\phantom{am}e}R\\_{cdef}\\nonumber \\\\\n & & \\hspace{4em}+2X\\phi\\_{a}\\phi^{e}\\left(R\\_{cdef}\\phi\\_{b}^{f}\\phi\\_{r}^{r}-R\\_{cdef}\\phi\\_{b}^{m}\\phi\\_{m}^{f}\\right)\\nonumber \\\\\n & & \\hspace{4em}+R\\_{cdef}\\phi\\_{a}\\phi^{e}\\phi^{n}\\phi^{m}\\left(-\\phi\\_{m}^{f}\\phi\\_{nb}+\\phi\\_{b}^{f}\\phi\\_{mn}\\right)\\Big\\}\\nonumber \\\\\n & \\equiv & -\\frac{1}{2}\\left(F\\_{4}^{(2;0,0)}+F\\_{5}^{(2;0,0)}+F\\_{2}^{(1;2,0)}+F\\_{7}^{(1;2,0)}-F\\_{9}^{(1;2,0)}-F\\_{10}^{(1;2,0)}\\right),\\qquad \\end{aligned}$$ \nand\n \n$$\\begin{aligned}\n \\varepsilon\\_{ijk}KK\\_{l}^{i}\\nabla^{j}K^{kl} & \\rightarrow & \\varepsilon^{abcd}u\\_{a}KK\\_{b}^{e}\\mathrm{D}\\_{c}K\\_{de}\\nonumber \\\\\n & = & -\\frac{1}{16X^{3}}\\varepsilon^{abcd}\\phi\\_{a}\\phi^{e}\\left(2R\\_{cdef}X\\phi\\_{b}^{f}\\phi\\_{r}^{r}+R\\_{cdrs}\\phi^{r}\\phi^{f}\\phi\\_{b}^{s}\\phi\\_{ef}\\right)\\nonumber \\\\\n & \\equiv & -\\frac{1}{2}\\left(F\\_{9}^{(1;2,0)}+F\\_{10}^{(1;2,0)}\\right). \\label{d4cor6}\\end{aligned}$$ \nThe last 2 monomials in the complete basis ([Z4]) involve *F* or $\\pounds\\_{\\bm{u}}K\\_{ij}$, which were not considered previously. Their covariant correspondences are\n \n$$\\begin{aligned}\n \\varepsilon\\_{ijk}FK\\_{l}^{i}\\nabla^{j}K^{kl} & \\rightarrow & \\varepsilon^{abcd}u\\_{a}FK\\_{fb}\\mathrm{D}\\_{c}K\\_{d}^{f}\\nonumber \\\\\n & = & -\\frac{1}{16X^{3}}\\varepsilon^{abcd}R\\_{cdrs}\\phi\\_{a}\\phi^{r}\\phi\\_{b}^{s}\\phi\\_{fe}\\phi^{e}\\phi^{f}\\nonumber \\\\\n & \\equiv & -\\frac{1}{2}F\\_{10}^{(1;2,0)}, \\label{d4cor7}\\end{aligned}$$ \nand\n \n$$\\begin{aligned}\n \\varepsilon\\_{ijk}a^{i}K^{jl}\\pounds\\_{\\bm{u}}K\\_{l}^{k} & \\rightarrow & \\varepsilon^{abcd}u\\_{a}a\\_{b}K\\_{c}^{f}h\\_{d}^{e}\\pounds\\_{\\bm{u}}K\\_{ef}\\nonumber \\\\\n & = & \\frac{1}{8X^{3}}\\varepsilon^{abcd}\\phi\\_{a}\\phi^{e}\\phi^{f}\\phi\\_{eb}\\left(\\phi\\_{drf}\\phi\\_{c}^{r}+R\\_{dfrs}\\phi^{r}\\phi\\_{c}^{s}\\right)\\nonumber \\\\\n & \\equiv & F\\_{6}^{(0;2,1)}-F\\_{8}^{(1;2,0)}\\nonumber \\\\\n & \\equiv & -\\frac{1}{2}F\\_{7}^{(1;2,0)}-F\\_{8}^{(1;2,0)}+\\nabla\\_{d}\\left(\\frac{1}{\\sigma^{6}}\\varepsilon\\_{abc}^{\\phantom{abc}d}\\phi^{e}\\phi^{f}\\phi^{a}\\phi\\_{e}^{b}\\phi^{cm}\\phi\\_{mf}\\right), \\label{d4cor8}\\end{aligned}$$ \nwhere again we have used ([app:F0216red]). We emphasize that the [*F**K*∇*K*] term and $[aK\\pounds K]$ are independent in the sense that they cannot be reduced or related to each other by integration by parts.\nTo summarize, we find 7 combinations of GST monomials, given in ([d3cor]) and ([d4cor1])-([d4cor6]), which are free of the Ostrogradsky ghost in the unitary gauge. On the other hand, the 2 combinations ([d4cor7]) and ([d4cor8]) contain linear terms in either $\\pounds\\_{\\bm{u}}K\\_{ij}$ or *F*, which are thus potentially risky even in the unitary gauge.\nIt is convenient to further combine ([d3cor])-([d4cor6]) to obtain a set of 7 independent, ghost-free parity-violating Lagrangians. For *d* = 3, we choose\n \n$$\\mathcal{L}\\_{1}\\equiv F\\_{1}^{(1;1,0)}=\\frac{1}{\\sigma^{3}}\\varepsilon\\_{abcd}R\\_{ef}^{\\phantom{ef}cd}\\phi^{a}\\phi^{e}\\phi^{bf},\\label{Lag1}$$ \nwhich is, in fact, the single GST monomial of the (1; 1, 0) category, i.e., linear in both the curvature tensor and the second-order derivative of the scalar field. For *d* = 4, there are 5 combinations of the (1; 2, 0) category, i.e., linear in the curvature tensor and quadratic in the second derivative of the scalar field, which we choose to be\n \n$$\\begin{aligned}\n \\mathcal{L}\\_{2} & \\equiv & F\\_{2}^{(1;2,0)}=\\frac{1}{\\sigma^{4}}\\varepsilon\\_{abcd}R\\_{ef}^{\\phantom{ef}cd}\\phi^{a}\\phi^{e}\\phi\\_{m}^{b}\\phi^{fm},\n \\end{aligned}$$ \n \n$$\\mathcal{L}\\_{3}\\equiv F\\_{7}^{(1;2,0)}=\\frac{1}{\\sigma^{6}}\\varepsilon\\_{abcd}R\\_{ef}^{\\phantom{ef}cd}\\phi^{m}\\phi^{n}\\phi^{e}\\phi^{a}\\phi\\_{m}^{f}\\phi\\_{n}^{b},$$ \n \n$$\\begin{aligned}\n \\mathcal{L}\\_{4} & \\equiv & F\\_{6}^{(1;2,0)}+F\\_{8}^{(1;2,0)}\\nonumber \\\\\n & = & \\frac{1}{\\sigma^{4}}\\varepsilon\\_{abcd}R^{amen}\\phi^{b}\\phi^{f}\\phi\\_{e}^{c}\\phi\\_{f}^{d}\\left(g\\_{mn}+\\frac{1}{2X}\\phi\\_{n}\\phi\\_{m}\\right),\n \\end{aligned}$$ \n \n$$\\begin{aligned}\n \\mathcal{L}\\_{5} & \\equiv & F\\_{5}^{(1;2,0)}-F\\_{8}^{(1;2,0)}\\nonumber \\\\\n & = & \\frac{1}{\\sigma^{4}}\\varepsilon\\_{abcd}R\\_{ef}^{\\phantom{ef}cm}\\phi^{a}\\phi^{e}\\phi^{bf}\\phi^{dn}\\left(g\\_{mn}+\\frac{1}{2X}\\phi\\_{m}\\phi\\_{n}\\right),\n \\end{aligned}$$ \n \n$$\\begin{aligned}\n \\mathcal{L}\\_{6} & \\equiv & F\\_{9}^{(1;2,0)}+F\\_{10}^{(1;2,0)}\\nonumber \\\\\n & = & \\frac{1}{\\sigma^{4}}\\varepsilon\\_{abcd}R\\_{ef}^{\\phantom{ef}cd}\\phi^{a}\\phi^{e}\\phi^{bf}\\phi^{mn}\\left(g\\_{mn}+\\frac{1}{2X}\\phi\\_{m}\\phi\\_{n}\\right).\n \\end{aligned}$$ \nThere is also a combination of the (2; 0, 0) category, i.e., quadratic in the curvature tensor and without higher order derivatives of the scalar field, which we choose to be\n \n$$\\begin{aligned}\n \\mathcal{L}\\_{7} & \\equiv & F\\_{4}^{(2;0,0)}+F\\_{5}^{(2;0,0)}\\nonumber \\\\\n & = & \\frac{1}{\\sigma^{2}}\\varepsilon\\_{abcd}R\\_{ef}^{\\phantom{ef}cd}R^{amen}\\phi^{b}\\phi^{f}\\left(g\\_{mn}+\\frac{1}{2X}\\phi\\_{m}\\phi\\_{n}\\right). \\label{Lag7}\n \\end{aligned}$$ \nThe 7 Lagrangians ([Lag1])-([Lag7]) are the main results in this work. Due to their generality and importance, we dub them the “Qi-Xiu” Lagrangians for the sake of brevity[9](#fn9).\nRecall that we have in total 17 GST monomials of *d* = 4 (see Table [tab:gstmono]). According to ([app:F0216red]) and ([app:F1014red]), *F*6(0; 2, 1) and *F*4(1; 0, 1) can be reduced by integrations by parts. Moreover, there are 7 identities in the unitary gauge among the rest 15 GST monomials of categories (1; 2, 0) and (2; 0, 0), which are shown in ([app:ugid1])-([app:ugid7]). As a result, besides the 2 potentially dangerous terms, we are left with exactly 7 independent GST terms that are ghost-free in the unitary gauge. We thus conclude that up to *d* = 4, any parity-violating scalar-tensor theory that is ghost-free in the unitary gauge can be expressed as\n \nL = ∑*a* = 17*C**a*(*ϕ*, *X*)L*a*, \nthat is, linear combinations of the “Qi-Xiu” Lagrangians with coefficients *C**a* being general functions of *ϕ* and *X*.\nComparing with the existing theories\n------------------------------------\nAccording to the previous section, up to *d* = 4, there are, in total, 9 independent SCG monomials, of which 7 contain no *F* or $\\pounds\\_{\\bm{u}}K\\_{ij}$ terms and thus are automatically healthy, and 2 are linear in *F* or $\\pounds\\_{\\bm{u}}K\\_{ij}$ and thus are dangerous. Their generally covariant correspondence is 9 GST polynomials. Any parity-violating GST polynomial up to *d* = 4 can be expressed as a linear combination of these 9 combinations up to the identities in the unitary gauge ([app:ugid1])-([app:ugid7]).\nLet us take Chern-Simons gravity as an example. The usual Chern-Simons term corresponds to the monomial *F*1(2; 0, 0) in our classification of GST monomials. The “covariant” 3+1 decomposition of Chern-Simons gravity with coefficient *f*(*ϕ*, *X*) is given by\n \n$$\\begin{aligned}\n \\mathcal{L}\\_{\\mathrm{CS}} & = &f(\\phi,X)F\\_{1}^{(2;0,0)} \\nonumber\\\\ \n & = & f(\\phi,X)n^{a}\\varepsilon\\_{abcd}\\Big(-2\\mathrm{D}^{e}a^{b}\\mathrm{D}^{c}K\\_{e}^{d}+2\\mathrm{D}^{e}K^{fb}K\\_{e}^{c}K\\_{f}^{d}-2\\mathrm{D}^{b}K^{fc}K\\_{f}^{p}K\\_{p}^{d} \\nonumber \\\\ \n & &\\qquad\\qquad\\qquad +{}^{3}\\!R\\_{ef}^{\\phantom{ef} cd}\\mathrm{D}^{e}K^{fb}+2h^{pd}\\mathrm{D}^{b}K^{fc}\\pounds\\_{\\bm{n}}K\\_{fp}\\Big), \\label{CSug}\n \\end{aligned}$$ \nwhere the covariant 3+1 decomposition is performed with respect to an arbitrary normal vector *n**a* (with no relation to *ϕ*), i.e., no specific gauge is taken. Ostrogradsky ghosts would appear in this case since the second-order time derivative of *h**i**j*, namely $\\pounds\\_{\\bm{n}}K\\_{ij}$, is kinetically mixed with the dynamical scalar field through the function of *X*, which cannot be reduced by integration by parts. Even in the unitary gauge with *n**a* = *u**a* ≡ − *N*∇*a**ϕ*, the risky term $\\pounds\\_{\\bm{u}}K\\_{ij}$ will still be present, although it only appears linearly in the Lagrangian. This is consistent with the previous analysis since *F*1(2; 0, 0) is not ghost-free by itself.\nAs a special case, if the coefficient *f* is a function of *ϕ* only, the term linear in $\\pounds\\_{\\bm{n}}K\\_{ij}$ can be reduced by integration by parts, which yields \n \n$$\\begin{aligned}\nf(\\phi)F\\_{1}^{(2;0,0)} & \\xlongequal{\\mathrm{u.g.}} & 8\\varepsilon^{ijk}f\\Big(K\\_{il}K^{lm}\\nabla\\_{j}K\\_{km}+K\\_{i}^{l}K\\_{j}^{m}\\nabla\\_{m}K\\_{kl}-KK\\_{i}^{l}\\nabla\\_{j}K\\_{kl}\\\\\n& & \\qquad-2\\,{}^{3}\\!R\\_{i}^{l}\\nabla\\_{j}K\\_{kl}-\\frac{1}{N}\\frac{\\dot{f}}{f}K\\_{i}^{l}\\nabla\\_{j}K\\_{kl}-\\frac{2}{N}\\nabla\\_{i}K\\_{jl}\\nabla\\_{k}\\nabla^{l}N\\Big), \\label{CS\\_fP\\_ug}\\end{aligned}$$ \nin the unitary gauge after fixing the spatial coordinates. It is transparent that in the unitary gauge, Chern-Simons gravity with coefficient *f*(*ϕ*) reduces to the form of a ghost-free SCG (i.e., with 3 DoFs). In particular, it takes the form of a linear combination of 6 monomials in ([Z3]) and ([Z4]).\nSimilar analysis can be performed for chiral scalar-tensor theories proposed in, in which three classes of Lagrangians without the Ostrogradsky ghosts in the unitary gauge were identified. The first class of Lagrangian is a linear combination of the following four terms:\n \nLPV1 = ∑*n* = 14*a**n*(*ϕ*, *X*)A*n*, \nwhere A1, ⋯, A4 correspond to (2; 0, 0) category in our notation[10](#fn10),\n \n$$\\begin{aligned}\n \\mathcal{A}\\_{1} & = & \\varepsilon^{abcd}R\\_{cdef}R\\_{ab\\phantom{e}g}^{\\phantom{ab}e}\\phi^{f}\\phi^{g}\\equiv-\\sigma^{2}F\\_{2}^{(2;0,0)}, \\label{calA1}\\\\\n \\mathcal{A}\\_{2} & = & \\varepsilon^{abcd}R\\_{cdef}R\\_{\\phantom{f}b}^{f}\\phi^{e}\\phi\\_{a}\\equiv\\sigma^{2}F\\_{4}^{(2;0,0)},\\\\\n \\mathcal{A}\\_{3} & = & \\varepsilon^{abcd}R\\_{cdef}R\\_{ag}^{\\phantom{ag}ef}\\phi\\_{b}\\phi^{g}\\equiv\\sigma^{2}F\\_{3}^{(2;0,0)},\\\\\n \\mathcal{A}\\_{4} & = & \\varepsilon^{abcd}R\\_{cdef}R\\_{\\phantom{ef}ab}^{ef}\\phi\\_{g}\\phi^{g}\\equiv-\\sigma^{2}F\\_{1}^{(2;0,0)}, \\label{calA4}\n \\end{aligned}$$ \nwhere *a*1, ⋯, *a*4 are general functions of *ϕ* and 2*X* ≡ − (∂*ϕ*)2 ≡ *σ*2. In the above we have explicitly denoted these terms in the notation of GST monomials *F**n*(2; , 0, 0). According to ([F2001dec])-([F2005dec]), in the unitary gauge, there are terms linear in $\\pounds\\_{\\bm{u}} K\\_{ij}$, which may be dangerous due to the second order time derivative encoded in $\\pounds\\_{\\bm{u}}K\\_{ij}$. The coefficient of the combination of such terms is proportional to 4*a*1 + 2*a*2 + *a*3 + 8*a*4. Therefore, the degenerate condition\n \n4*a*1 + *a*2 + 2*a*3 + 8*a*4 = 0, \nidentified in can be regarded as requiring the vanishing of the dangerous term $[\\pounds K\\nabla K]$ in the unitary gauge.\nSince the 4 coefficients *a*1, ⋯, *a*4 are subject to a single constraint ([pvmod1cond]), we have 3 combinations of terms:\n \n$$\\begin{aligned}\n \\mathcal{O}\\_{1} & \\equiv & \\mathcal{A}\\_{1}-\\frac{1}{2}\\mathcal{A}\\_{4}=\\varepsilon^{abcd}R\\_{cdef}\\left(R\\_{ab\\phantom{e}g}^{\\phantom{ab}e}\\phi^{f}\\phi^{g}-\\frac{1}{2}R\\_{\\phantom{ef}ab}^{ef}\\phi\\_{g}\\phi^{g}\\right), \\label{pv1O1}\\\\\n \\mathcal{O}\\_{2} & \\equiv & \\mathcal{A}\\_{2}-\\frac{1}{8}\\mathcal{A}\\_{4}=\\varepsilon^{abcd}R\\_{cdef}\\left(R\\_{\\phantom{f}b}^{f}\\phi^{e}\\phi\\_{a}-\\frac{1}{8}R\\_{\\phantom{ef}ab}^{ef}\\phi\\_{g}\\phi^{g}\\right), \\label{pv1O2}\\\\\n \\mathcal{O}\\_{3} & \\equiv & \\mathcal{A}\\_{3}-\\frac{1}{4}\\mathcal{A}\\_{4}=\\varepsilon^{abcd}R\\_{cdef}\\left(R\\_{ag}^{\\phantom{ag}ef}\\phi\\_{b}\\phi^{g}-\\frac{1}{4}R\\_{\\phantom{ef}ab}^{ef}\\phi\\_{g}\\phi^{g}\\right), \\label{LPVc1}\n \\end{aligned}$$ \nwhich are free of the Ostrogradsky ghost in the unitary gauge. Indeed, in the unitary gauge one finds that \n \n$$\\mathcal{O}\\_{1}^{\\mathrm{(u.g.)}}=-\\frac{8}{N^{2}}\\varepsilon\\_{ijk}K^{li}K^{mj}\\nabla\\_{m}K\\_{l}^{\\phantom{l}k}+\\frac{8}{N^{2}}\\varepsilon\\_{ijk}\\,{}^{3}\\!R^{li}\\nabla^{k}K\\_{l}^{\\phantom{l}j},$$ \n \n$$\\begin{aligned}\n \\mathcal{O}\\_{2}^{\\mathrm{(u.g.)}} & = & -\\frac{2}{N^{2}}\\varepsilon\\_{ijk}K^{li}K^{mj}\\nabla\\_{m}K\\_{l}^{\\phantom{l}k}-\\frac{2}{N^{2}}\\varepsilon\\_{ijk}\\left(K\\_{m}^{\\phantom{l}i}K^{lm}-KK^{li}\\right)\\nabla^{k}K\\_{l}^{\\phantom{l}j}\\nonumber \\\\\n & & +\\frac{4}{N^{2}}\\varepsilon\\_{ijk}\\,{}^{3}\\!R^{li}\\nabla^{k}K\\_{l}^{\\phantom{l}j},\n \\end{aligned}$$ \nand O3(u.g.) ≡ 0[11](#fn11). Therefore, when restricted to the 4 GST monomials in ([calA1])-([calA4]), there are 2 independent ghost-free combinations, given in ([pv1O1]) and ([pv1O2]).\nThe second class of Lagrangian is composed of a single term belonging to the (1; 1, 0) category,\n \nLPV2 = *b*(*ϕ*, *X*)B, with B = *ɛ**a**b**c**d**R**c**d**e**f**ϕ**a**ϕ**e**ϕ**b**f* ≡ *σ*3*F*1(1; 1, 0), \nwhich is healthy in the unitary gauge since\n \n$$\\mathcal{B}^{\\mathrm{(u.g.)}} = \\frac{2}{N^{3}}\\varepsilon\\_{ijk}K^{li}\\nabla^{j}K\\_{l}^{\\phantom{l}k}.$$ \nThe third class of Lagrangian is the linear combination of 6 monomials belonging to the (1; 2, 0) category,\n \nLPV3 = ∑*n* = 16*c**n*(*ϕ*, *X*)C*n*, \nwith[12](#fn12)\n \n$$\\begin{aligned}\n \\mathcal{C}\\_{1} & = & \\varepsilon^{abcd}R\\_{cdef}\\phi\\_{a}^{e}\\phi\\_{b}^{f}\\phi\\_{g}\\phi^{g}\\equiv-\\sigma^{4}F\\_{1}^{(1;2,0)}, \\label{calC1}\\\\\n \\mathcal{C}\\_{2} & = & \\varepsilon^{abcd}R\\_{cdef}\\phi\\_{a}^{e}\\phi\\_{b}^{g}\\phi^{f}\\phi\\_{g}\\equiv\\sigma^{4}F\\_{3}^{(1;2,0)},\\\\\n \\mathcal{C}\\_{3} & = & \\varepsilon^{abcd}R\\_{cefg}\\phi\\_{a}^{f}\\phi\\_{b}^{g}\\phi^{e}\\phi\\_{d}\\equiv-2\\sigma^{4}F\\_{5}^{(1;2,0)},\\\\\n \\mathcal{C}\\_{4} & = & \\varepsilon^{abcd}R\\_{cdef}\\phi\\_{a}^{e}\\phi\\_{g}^{f}\\phi\\_{b}\\phi^{g}\\equiv-\\sigma^{4}F\\_{4}^{(1;2,0)},\\\\\n \\mathcal{C}\\_{5} & = & \\varepsilon^{abcd}R\\_{de}\\phi\\_{a}^{e}\\phi\\_{b}^{f}\\phi\\_{c}\\phi\\_{f}\\equiv-\\sigma^{4}F\\_{6}^{(1;2,0)},\\\\\n \\mathcal{C}\\_{6} & = & \\varepsilon^{abcd}R\\_{cdef}\\phi^{e}\\phi\\_{a}\\phi\\_{b}^{f}\\square\\phi\\equiv \\sigma^{4}F\\_{9}^{(1;2,0)}, \\label{calC6}\\end{aligned}$$ \nwhere *c*1, ⋯, *c*6 are general functions of *ϕ* and *X*. After some manipulations, one can show that the coefficients of $\\pounds\\_{\\bm{u}}K\\_{ij}$ and $\\pounds\\_{\\bm{u}}N$ are proportional to 4*c*1 + 2*c*2 + 2*c*3 − *c*5 and 2*c*1 + *c*2 + *c*4 + *c*6, respectively. In, *c*6 is set to be vanishing[13](#fn13). Thus one requires\n \n$$\\begin{aligned}\n 4c\\_{1}+2c\\_{2}+2c\\_{3}-c\\_{5} & = & 0,\\\\\n 2c\\_{1}+c\\_{2}+c\\_{4} & = & 0,\n \\end{aligned}$$ \nin order to evade the Ostrogradsky ghost. Now there are 5 coefficients *c*1, ⋯, *c*5 subject to 2 constraints. Therefore, we have 3 combinations of terms that are free of Ostrogradsky ghosts in the unitary gauge:\n \n$$\\begin{aligned}\n \\mathcal{O}\\_{1} & \\equiv & \\mathcal{C}\\_{1}-2\\mathcal{C}\\_{4}+4\\mathcal{C}\\_{5}\\nonumber \\\\\n & = & \\varepsilon^{abcd}\\left[R\\_{cdef}\\left(\\phi\\_{b}^{f}\\phi\\_{g}\\phi^{g}-2\\phi\\_{g}^{f}\\phi\\_{b}\\phi^{g}\\right)+4R\\_{de}\\phi\\_{b}^{f}\\phi\\_{c}\\phi\\_{f}\\right]\\phi\\_{a}^{e}, \\label{pv3O1}\n \\end{aligned}$$ \n \n$$\\begin{aligned}\n \\mathcal{O}\\_{2} & \\equiv & \\mathcal{C}\\_{2}-\\mathcal{C}\\_{4}+2\\mathcal{C}\\_{5}\\nonumber \\\\\n & = & \\varepsilon^{abcd}\\left[R\\_{cdef}\\left(\\phi\\_{b}^{g}\\phi^{f}\\phi\\_{g}-\\phi\\_{g}^{f}\\phi\\_{b}\\phi^{g}\\right)+2R\\_{de}\\phi\\_{b}^{f}\\phi\\_{c}\\phi\\_{f}\\right]\\phi\\_{a}^{e},\n \\end{aligned}$$ \nand\n \nO3 ≡ C3 + 2C5 = *ɛ**a**b**c**d*(*R**c**g**e**f**ϕ**g**ϕ**d* + 2*R**d**e**ϕ**c**ϕ**f*)*ϕ**a**e**ϕ**b**f*. \nThis can be checked explicitly. In the unitary gauge,\n \n$$\\begin{aligned}\n \\mathcal{O}\\_{1}^{\\mathrm{(u.g.)}} & = & \\frac{4}{N^{4}}\\varepsilon\\_{ijk}\\left(K^{li}K^{mj}\\nabla\\_{m}K\\_{l}^{\\phantom{b}k}-\\frac{1}{N}K^{lj}\\,{}^{3}\\!R\\_{l}^{\\phantom{l}k}\\nabla^{i}N\\right),\n \\end{aligned}$$ \nwhile O2(u.g.) = 0 and $\\mathcal{O}\\_{3}^{\\mathrm{(u.g.)}}=\\frac{1}{2}\\mathcal{L}\\_{1}^{\\mathrm{(u.g.)}}$. Therefore, when restricted to the 6 GST monomials in ([calC1])-([calC6]), there is only one independent ghost-free combination given in ([pv3O1]).\nConclusions\n===========\nRecently, there has been an increasing interest in studying gravitational theories with parity violation. In this work, we have investigated the scalar-tensor theory with parity violation. In particular, we are looking for the special combinations of the scalar-tensor monomials that are free of the Ostrogradsky ghost in the unitary gauge, i.e., when the scalar field possesses a timelike gradient.\nSince the generally covariant scalar-tensor theory (GST) in the unitary gauge takes the form of spatially covariant gravity (SCG), in Sec. [sec:scg], we describe the general framework of SCG theory and the classification of SCG monomials. We extend the SCG theory by introducing Lie derivatives of the lapse function $F\\equiv \\pounds\\_{\\bm{u}} \\ln N$ and the extrinsic curvature $\\pounds\\_{\\bm{u}}K\\_{ij}$. This is not only because they are the natural building blocks of SCG, but also because they necessarily arise in the decomposition of scalar-tensor theory in the unitary gauge up to *d* = 4 with *d* the total number of derivatives.\nIn Sec. [sec:pvscg], by including *F* and $\\pounds\\_{\\bm{u}}K\\_{ij}$ as the building blocks of SCG, we exhausted the SCG monomials up to *d* = 4, which are classified and summarized in Table [SCGpvmono]. Based on this classification, we obtain the complete basis for the parity-violating SCG polynomials of *d* = 3 and *d* = 4, which are given in ([Z3]) and ([Z4]), respectively. In total, there are 9 independent SCG monomials with parity violation, of which 7 contain no higher temporal derivatives and are thus automatically free of ghosts, while 2 involve Lie derivatives of the extrinsic curvature and the lapse function and are thus potentially risky. Our analysis thus generalizes the previous result presented in.\nOur final goal is to find the ghost-free combinations of GST monomials, which is performed in Sec. [sec:gfgst]. To this end, in Sec. [subsec:dec] we derive the decomposition of the GST monomials in the unitary gauge. The resulting expressions show the dangerous temporal derivative terms explicitly and thus give us guidance on finding the ghost-free combinations. The main results in this work are presented in Sec. [subsec:gfst], where we derive the generally covariant correspondence of the 9 parity-violating SCG monomials in the complete basis ([Z3]) and ([Z4]). Since 7 out of the 9 SCG monomials are ghost-free, there must be 7 scalar-tensor Lagrangians that are ghost-free in the unitary gauge, which we choose to be ([Lag1])-([Lag7]) and dub them the “Qi-Xiu” Lagrangians for short. Up to *d* = 4, since any parity-violating GST polynomial in the unitary gauge takes the form of a linear combination of the 9 parity-violating SCG monomials in the complete basis, we conclude that, up to 7 identities in the unitary gauge ([app:ugid1])-([app:ugid7]), the “Qi-Xiu” Lagrangians ([Lag1])-([Lag7]) are the most general parity-violating scalar-tensor theories that are ghost-free in the unitary gauge up to *d* = 4. As shown in Sec. [subsec:comp], our results have included the Chern-Simons term as well as the chiral scalar-tensor theory proposed in as special cases.\nWe thank Yi-Fu Cai, Yang Yu and Zhiyu Lu for valuable discussions. This work was partly supported by the National Natural Science Foundation of China (NSFC) under the grants No. 12347137 (YMH) and No. 11975020 (XG).\nReduction of $[\\pounds K \\nabla K]$ term\n========================================\nIn this appendix, we will show that $\\varepsilon\\_{ijk}\\pounds\\_{\\bm{u}}K\\_{l}^{i}\\nabla^{j}K^{kl}$ can be reduced by integration by parts. When performing integration by parts regarding the SCG terms, it is more convenient to use the generally covariant corresponding expressions. In our case, it is given by\n \n$$\\varepsilon^{ijk}\\pounds\\_{\\bm{u}}K\\_{li}\\nabla\\_{j}K\\_{k}^{l}\\rightarrow f\\varepsilon^{\\bm{u}\\hat{b}\\hat{c}\\hat{d}}h^{ef}\\pounds\\_{\\bm{u}}K\\_{be}\\mathrm{D}\\_{c}K\\_{df} \\equiv \\mathcal{O},$$ \nwhere we define $\\varepsilon^{\\bm{u}\\hat{b}\\hat{c}\\hat{d}}\\equiv u\\_{a}h\\_{b'}^{b}h\\_{c'}^{c}h\\_{d'}^{d}\\varepsilon^{ab'c'd'}$ for later convenience.\nThere are two equivalent approaches to performing the integrations by parts: either using the Lie derivative and intrinsic derivative directly, or using the 4-dimensional covariant derivative ∇*a* by expanding Lie/intrinsic derivatives in terms of the covariant derivative. Here we choose the former. We have\n \n$$\\begin{aligned}\n \\mathcal{O} & = & f\\varepsilon^{\\bm{u}\\hat{b}\\hat{c}\\hat{d}}h^{ef}\\pounds\\_{\\bm{u}}K\\_{be}\\mathrm{D}\\_{c}K\\_{df}\\nonumber \\\\\n & = & \\pounds\\_{\\bm{u}}\\left(f\\varepsilon^{\\bm{u}\\hat{b}\\hat{c}\\hat{d}}h^{ef}K\\_{be}\\mathrm{D}\\_{c}K\\_{df}\\right)-K\\_{be}\\pounds\\_{\\bm{u}}\\left(f\\varepsilon^{\\bm{u}\\hat{b}\\hat{c}\\hat{d}}h^{ef}\\mathrm{D}\\_{c}K\\_{df}\\right)\\nonumber \\\\\n & \\simeq & -Kf\\varepsilon^{\\bm{u}\\hat{b}\\hat{c}\\hat{d}}K\\_{b}^{f}\\mathrm{D}\\_{c}K\\_{df}-K\\_{be}\\pounds\\_{\\bm{u}}\\left(f\\varepsilon^{\\bm{u}\\hat{b}\\hat{c}\\hat{d}}h^{ef}\\right)\\mathrm{D}\\_{c}K\\_{df}-K\\_{be}f\\varepsilon^{\\bm{u}\\hat{b}\\hat{c}\\hat{d}}h^{ef}\\pounds\\_{\\bm{u}}\\mathrm{D}\\_{c}K\\_{df}\\nonumber \\\\\n & = & -f\\varepsilon^{\\bm{u}\\hat{b}\\hat{c}\\hat{d}}KK\\_{b}^{f}\\mathrm{D}\\_{c}K\\_{df}-K\\_{be}\\pounds\\_{\\bm{u}}\\left(f\\varepsilon^{\\bm{u}\\hat{b}\\hat{c}\\hat{d}}h^{ef}\\right)\\mathrm{D}\\_{c}K\\_{df}-K\\_{b}^{f}f\\varepsilon^{\\bm{u}\\hat{b}\\hat{c}\\hat{d}}\\left[\\pounds\\_{\\bm{u}},\\mathrm{D}\\_{c}\\right]K\\_{df}\\nonumber \\\\\n & & -\\mathrm{D}\\_{c}\\left(K\\_{b}^{f}f\\varepsilon^{\\bm{u}\\hat{b}\\hat{c}\\hat{d}}\\pounds\\_{\\bm{u}}K\\_{df}\\right)+\\mathrm{D}\\_{c}\\left(f\\varepsilon^{\\bm{u}\\hat{b}\\hat{c}\\hat{d}}\\right)K\\_{b}^{f}\\pounds\\_{\\bm{u}}K\\_{df}+\\mathrm{D}\\_{c}K\\_{b}^{f}f\\varepsilon^{\\bm{u}\\hat{b}\\hat{c}\\hat{d}}\\pounds\\_{\\bm{u}}K\\_{df},\\label{calOint}\\end{aligned}$$ \nwhere we have used that for a scalar field Φ, $\\pounds\\_{\\bm{u}}\\Phi=u^{a}\\nabla\\_{a}\\Phi\\simeq-K\\Phi$, and the commutator is defined by\n \n$$\\left[\\pounds\\_{\\bm{u}},\\mathrm{D}\\_{c}\\right]K\\_{df}=\\pounds\\_{\\bm{u}}\\left(\\mathrm{D}\\_{c}K\\_{df}\\right)-\\mathrm{D}\\_{c}\\left(\\pounds\\_{\\bm{u}}K\\_{df}\\right).$$ \nFor the second term in ([calOint]), by using \n$$\\pounds\\_{\\bm{u}}f=\\frac{\\partial f}{\\partial\\phi}\\pounds\\_{\\bm{u}}\\phi+N\\frac{\\partial f}{\\partial N}F,$$ \n with $F\\equiv\\frac{1}{N}\\pounds\\_{\\bm{u}}N$, \n$$\\pounds\\_{\\bm{u}}h^{ab}=-2K^{ab}+2u^{(a}a^{b)},$$ \n and \n$$\\pounds\\_{\\bm{u}}\\varepsilon^{\\bm{n}\\hat{a}\\hat{b}\\hat{c}}=a\\_{d}\\varepsilon^{d\\hat{a}\\hat{b}\\hat{c}}-\\left(K\\_{e}^{a}-u^{a}a\\_{e}\\right)\\varepsilon^{\\bm{u}e\\hat{b}\\hat{c}}-\\left(K\\_{e}^{b}-u^{b}a\\_{e}\\right)\\varepsilon^{\\bm{u}\\hat{a}e\\hat{c}}-\\left(K\\_{e}^{c}-u^{c}a\\_{e}\\right)\\varepsilon^{\\bm{u}\\hat{a}\\hat{b}e},$$ \n we find\n \n$$\\begin{aligned}\n K\\_{be}\\pounds\\_{\\bm{u}}\\left(f\\varepsilon^{\\bm{u}\\hat{b}\\hat{c}\\hat{d}}h^{ef}\\right)\\mathrm{D}\\_{c}K\\_{df} & = & \\left(\\frac{\\partial f}{\\partial\\phi}\\pounds\\_{\\bm{u}}\\phi+N\\frac{\\partial f}{\\partial N}F\\right)\\varepsilon^{\\bm{u}\\hat{b}\\hat{c}\\hat{d}}K\\_{b}^{f}\\mathrm{D}\\_{c}K\\_{df}\\nonumber \\\\\n & & -3f\\varepsilon^{\\bm{u}\\hat{b}\\hat{c}\\hat{d}}K\\_{m}^{f}K\\_{b}^{m}\\mathrm{D}\\_{c}K\\_{df}-f\\varepsilon^{\\bm{u}\\hat{b}\\hat{c}\\hat{d}}K\\_{b}^{f}K\\_{c}^{e}\\mathrm{D}\\_{e}K\\_{df}.\\end{aligned}$$ \nFor the third term in ([calOint]), the commutator is given by \n$$\\left[\\pounds\\_{\\bm{u}},\\mathrm{D}\\_{c}\\right]K\\_{df}=a\\_{c}\\pounds\\_{\\bm{u}}K\\_{df}+\\Xi\\_{cdm}K\\_{\\phantom{m}f}^{m}+\\Xi\\_{cfm}K\\_{d}^{\\phantom{d}m},$$ \n with \nΞ*c**d**m* = − (*a**c**K**d**m* + *a**d**K**m**c* − *a**m**K**c**d*) − (D*c**K**d**m* + D*d**K**m**c* − D*m**K**c**d*), \n and Ξ*c**f**m* is given accordingly. Thus\n \n$$K\\_{b}^{f}f\\varepsilon^{\\bm{u}\\hat{b}\\hat{c}\\hat{d}}\\left[\\pounds\\_{\\bm{u}},\\mathrm{D}\\_{c}\\right]K\\_{df}=-f\\varepsilon^{\\bm{u}\\hat{b}\\hat{c}\\hat{d}}a\\_{b}K\\_{c}^{f}\\pounds\\_{\\bm{u}}K\\_{df}-2f\\varepsilon^{\\bm{u}\\hat{b}\\hat{c}\\hat{d}}K\\_{b}^{f}K\\_{c}^{e}\\mathrm{D}\\_{e}K\\_{df}.$$ \nFor the fourth term in ([calOint]), we have\n \n$$\\mathrm{D}\\_{c}\\left(K\\_{b}^{f}f\\varepsilon^{\\bm{u}\\hat{b}\\hat{c}\\hat{d}}\\pounds\\_{\\bm{u}}K\\_{df}\\right)\\simeq f\\varepsilon^{\\bm{u}\\hat{b}\\hat{c}\\hat{d}}a\\_{b}K\\_{c}^{f}\\pounds\\_{\\bm{u}}K\\_{df},$$ \nwhere we have used that for a tangent vector *A**a*, D*a**A**a* = ∇*a**A**a* − *a**a**A**a*.\nFor the fifth term in ([calOint]), we have\n \n$$\\mathrm{D}\\_{c}\\left(f\\varepsilon^{\\bm{u}\\hat{b}\\hat{c}\\hat{d}}\\right)K\\_{b}^{f}\\pounds\\_{\\bm{u}}K\\_{df}=-\\varepsilon^{\\bm{u}\\hat{b}\\hat{c}\\hat{d}}N\\frac{\\partial f}{\\partial N}a\\_{b}K\\_{c}^{f}\\pounds\\_{\\bm{u}}K\\_{df},$$ \nwhere we have used $\\mathrm{D}\\_{c}\\varepsilon^{\\bm{u}\\hat{b}\\hat{c}\\hat{d}}=K\\_{ce}\\varepsilon^{e\\hat{b}\\hat{c}\\hat{d}}$.\nPutting all the above together, and noting that the last term in ([calOint]) is nothing but − O, we get\n \n$$\\begin{aligned}\n \\mathcal{O} & \\simeq & -\\frac{1}{2}f\\varepsilon^{\\bm{u}\\hat{b}\\hat{c}\\hat{d}}KK\\_{b}^{f}\\mathrm{D}\\_{c}K\\_{df}+\\frac{3}{2}f\\varepsilon^{\\bm{n}\\hat{b}\\hat{c}\\hat{d}}K\\_{m}^{f}K\\_{b}^{m}\\mathrm{D}\\_{c}K\\_{df}+\\frac{3}{2}f\\varepsilon^{\\bm{u}\\hat{b}\\hat{c}\\hat{d}}K\\_{b}^{f}K\\_{c}^{e}\\mathrm{D}\\_{e}K\\_{df}\\nonumber \\\\\n & & -\\frac{1}{2}\\left(\\frac{\\partial f}{\\partial\\phi}\\pounds\\_{\\bm{u}}\\phi+N\\frac{\\partial f}{\\partial N}F\\right)\\varepsilon^{\\bm{u}\\hat{b}\\hat{c}\\hat{d}}K\\_{b}^{f}\\mathrm{D}\\_{c}K\\_{df}\\nonumber \\\\\n & & -\\frac{1}{2}\\varepsilon^{\\bm{u}\\hat{b}\\hat{c}\\hat{d}}N\\frac{\\partial f}{\\partial N}a\\_{b}K\\_{c}^{f}\\pounds\\_{\\bm{u}}K\\_{df}.\\end{aligned}$$ \nIn terms of spatial indices with *t* = *ϕ*,\n \n$$\\begin{aligned}\n \\mathcal{O} & \\simeq & -\\frac{1}{2}f\\varepsilon^{ijk}KK\\_{i}^{l}\\nabla\\_{j}K\\_{kl}+\\frac{3}{2}f\\varepsilon^{ijk}K\\_{m}^{l}K\\_{i}^{m}\\nabla\\_{j}K\\_{kl}+\\frac{3}{2}f\\varepsilon^{ijk}K\\_{i}^{l}K\\_{j}^{m}\\nabla\\_{m}K\\_{kl}\\nonumber \\\\\n & & -\\frac{1}{2}\\left(\\frac{\\partial f}{\\partial t}+N\\frac{\\partial f}{\\partial N}F\\right)\\varepsilon^{ijk}K\\_{i}^{l}\\nabla\\_{j}K\\_{kl}-\\frac{1}{2}\\varepsilon^{ijk}N\\frac{\\partial f}{\\partial N}a\\_{i}K\\_{j}^{l}\\pounds\\_{\\bm{u}}K\\_{kl}.\\end{aligned}$$ \nTo conclude, we have explicitly shown that $\\varepsilon^{ijk}\\pounds\\_{\\bm{u}}K\\_{li}\\nabla\\_{j}K\\_{k}^{l}$ is not independent, which can be reduced to a linear combination of [*K**K*∇*K*], [*K*∇*K*], and $\\left[aK\\pounds K\\right]$ terms by integrations by parts.\nIntegrations by parts\n=====================\nFor the (1; 2, 0) category, the decomposition of the following GST monomials can be recast by integration by parts:\n \n$$\\begin{aligned}\n F\\_{1}^{(1;2,0)} & \\xlongequal{\\text{u.g.}} & 4\\varepsilon\\_{abc}\\Big(-\\,{}^{3}\\!R\\_{d}^{a}K^{db}a^{c}-2K\\_{d}^{a}a^{b}\\mathrm{D}^{c}a^{d}-FK^{ad}\\mathrm{D}^{b}K\\_{d}^{c}-K^{ad}K^{be}\\mathrm{D}\\_{e}K\\_{d}^{c}\\nonumber \\\\\n & & \\qquad-a^{a}K^{db}h^{ec}\\pounds\\_{\\boldsymbol{u}}K\\_{de}\\Big)-4\\nabla^{c}\\left(\\varepsilon\\_{abc}a^{a}a^{d}K\\_{d}^{b}\\right), \\label{F1201ibp}\\end{aligned}$$ \n \n$$F\\_{2}^{(1;2,0)} \\xlongequal{\\text{u.g.}} -2\\varepsilon\\_{abc}\\left(K\\_{e}^{a}a^{b}\\mathrm{D}^{c}a^{e}+K\\_{d}^{a}K^{de}\\mathrm{D}^{b}K\\_{e}^{c}\\right)-2\\nabla^{c}\\left(\\varepsilon\\_{abc}a^{a}a^{d}K\\_{d}^{b}\\right), \\label{F1202ibp}$$ \n \n$$\\begin{aligned}\n F\\_{3}^{(1;2,0)} & \\xlongequal{\\text{u.g.}} & 2\\varepsilon\\_{abc}\\left(2K\\_{d}^{a}a^{b}\\mathrm{D}^{c}a^{d}+FK^{ad}\\mathrm{D}^{b}K\\_{d}^{c}+a^{a}K^{db}h^{ec}\\pounds\\_{\\boldsymbol{u}}K\\_{de}\\right)\\nonumber \\\\\n & & +2\\nabla^{c}\\left(\\varepsilon\\_{abc}a^{a}a^{d}K\\_{d}^{b}\\right), \\label{F1203ibp}\\end{aligned}$$ \n \n$$F\\_{4}^{(1;2,0)} \\xlongequal{\\text{u.g.}} -2\\varepsilon\\_{abc}\\left(K\\_{e}^{a}a^{b}\\mathrm{D}^{c}a^{e}+\\,{}^{3}\\!R\\_{d}^{a}K^{db}a^{c}+FK^{ad}\\mathrm{D}^{b}K\\_{d}^{c}\\right)-2\\nabla^{c}\\left(\\varepsilon\\_{abc}a^{a}a^{d}K\\_{d}^{b}\\right),\\qquad \\label{F1204ibp}$$ \nand\n \n$$F\\_{7}^{(1;2,0)}\\xlongequal{\\text{u.g.}}2\\varepsilon\\_{abc}K\\_{e}^{a}a^{b}\\mathrm{D}^{c}a^{e}+2\\nabla^{c}\\left(\\varepsilon\\_{abc}a^{a}a^{d}K\\_{d}^{b}\\right). \\label{F1207ibp}$$ \nFor the (2; 0, 0) category, the decomposition of the following GST monomials can be recast by integration by parts:\n \n$$\\begin{aligned}\n F\\_{1}^{(2;0,0)} & \\xlongequal{\\text{u.g.}} & 8\\varepsilon\\_{abc}\\Big(2\\,{}^{3}\\!R^{ad}K\\_{d}^{b}a^{c}-2\\,{}^{3}\\!R^{da}\\mathrm{D}^{b}K\\_{d}^{c}+K\\_{d}^{a}K^{de}\\mathrm{D}^{b}K\\_{e}^{c}+K^{da}K^{eb}\\mathrm{D}\\_{e}K\\_{d}^{c}\\nonumber \\\\\n & & \\qquad-KK\\_{e}^{a}\\mathrm{D}^{b}K^{ce}\\Big)+8\\mathcal{D},\\label{F2001ibp}\n \\end{aligned}$$ \n \n$$\\begin{aligned}\n F\\_{2}^{(2;0,0)} & \\xlongequal{\\text{u.g.}} & 4\\varepsilon\\_{abc}\\Big(2\\,{}^{3}\\!R^{ad}K\\_{d}^{b}a^{c}+K\\_{d}^{a}K^{de}\\mathrm{D}^{b}K\\_{e}^{c}+3K^{da}K^{eb}\\mathrm{D}\\_{e}K\\_{d}^{c}\\nonumber \\\\\n & & \\qquad-KK\\_{e}^{a}\\mathrm{D}^{b}K^{ce}\\Big)+4\\mathcal{D},\\label{F2002ibp}\n \\end{aligned}$$ \n \n$$\\begin{aligned}\n F\\_{3}^{(2;0,0)} & \\xlongequal{\\text{u.g.}} & 2\\varepsilon\\_{abc}\\Big(-2\\,{}^{3}\\!R^{ad}K\\_{d}^{b}a^{c}+2\\,{}^{3}\\!R^{da}\\mathrm{D}^{b}K\\_{d}^{c}-K\\_{d}^{a}K^{de}\\mathrm{D}^{b}K\\_{e}^{c}-K^{da}K^{eb}\\mathrm{D}\\_{e}K\\_{d}^{c}\\nonumber \\\\\n & & \\qquad+KK\\_{e}^{a}\\mathrm{D}^{b}K^{ce}\\Big)-2\\mathcal{D},\\label{F2003ibp}\n \\end{aligned}$$ \n \n$$\\begin{aligned}\n F\\_{4}^{(2;0,0)} & \\xlongequal{\\text{u.g.}} & \\varepsilon\\_{abc}\\Big(-2\\,{}^{3}\\!R^{ad}K\\_{d}^{b}a^{c}-2\\,{}^{3}\\!R^{da}\\mathrm{D}^{b}K\\_{d}^{c}+K\\_{d}^{a}K^{de}\\mathrm{D}^{b}K\\_{e}^{c}-3K^{da}K^{eb}\\mathrm{D}\\_{e}K\\_{d}^{c}\\nonumber \\\\\n & & \\qquad-KK^{ea}\\mathrm{D}^{c}K\\_{e}^{b}\\Big)-\\mathcal{D},\\label{F2004ibp}\n \\end{aligned}$$ \nand\n \n$$\\begin{aligned}\n F\\_{5}^{(2;0,0)} & \\xlongequal{\\text{u.g.}} & \\varepsilon\\_{abc}\\Big(2\\,{}^{3}\\!R^{ad}K\\_{d}^{b}a^{c}+K\\_{d}^{a}K^{de}\\mathrm{D}^{b}K\\_{e}^{c}+3K^{da}K^{eb}\\mathrm{D}\\_{e}K\\_{d}^{c}\\nonumber \\\\\n & & \\qquad-KK^{ea}\\mathrm{D}^{b}K\\_{e}^{c}\\Big)+\\mathcal{D},\\label{F2005ibp}\n \\end{aligned}$$ \nwhere in ([F2001ibp])-([F2005ibp]), D stands for the total derivative\n \n$$\\begin{aligned}\n \\mathcal{D} & = & 2\\nabla^{c}\\left(\\varepsilon\\_{abc}\\left(\\mathrm{D}^{a}a^{d}+a^{a}a^{d}\\right)K\\_{d}^{b}\\right)\\nonumber \\\\\n & & +\\nabla\\_{e}\\left(n^{a}\\varepsilon\\_{abcd}n^{e}K\\_{f}^{d}\\nabla^{b}K^{cf}\\right)+\\nabla^{b}\\left(n^{a}\\varepsilon\\_{abcd}K^{cf}n^{e}\\nabla\\_{e}K\\_{f}^{d}\\right). \\label{app:calD}\n \\end{aligned}$$ \nIn deriving the decomposition in the unitary gauge, we frequently make use of the following integrations by parts\n \n$$\\begin{aligned}\n \\varepsilon\\_{abc}\\mathrm{D}^{a}a^{d}\\mathrm{D}^{c}K\\_{d}^{b} & = & -\\varepsilon\\_{abc}\\left(K\\_{e}^{a}a^{b}\\mathrm{D}^{c}a^{e}-\\,{}^{3}\\!R\\_{e}^{a}K^{be}a^{c}\\right)+\\nabla^{c}\\left(\\varepsilon\\_{abc}\\mathrm{D}^{a}a^{d}K\\_{d}^{b}\\right),\\\\\n \\varepsilon\\_{abc}a^{a}a^{d}\\mathrm{D}^{c}K\\_{d}^{b} & = & \\varepsilon\\_{abc}K\\_{e}^{a}a^{b}\\mathrm{D}^{c}a^{e}+\\nabla^{c}\\left(\\varepsilon\\_{abc}a^{a}a^{d}K\\_{d}^{b}\\right),\\end{aligned}$$ \nand\n \n$$\\begin{aligned}\n \\varepsilon^{abcd}u\\_{a}\\pounds\\_{\\bm{u}}K\\_{be}\\mathrm{D}\\_{c}K\\_{d}^{e} & = & \\frac{1}{2}\\varepsilon\\_{abc}\\left(3K\\_{e}^{a}K\\_{f}^{e}\\mathrm{D}^{b}K^{cf}+3K^{ea}K^{rb}\\mathrm{D}\\_{r}K\\_{e}^{c}-KK\\_{f}^{a}\\mathrm{D}^{b}K^{cf}\\right)\\nonumber \\\\\n & & +\\frac{1}{2}\\nabla\\_{e}\\left(\\varepsilon\\_{abc}u^{e}K\\_{f}^{c}\\nabla^{a}K^{bf}\\right)+\\frac{1}{2}\\nabla^{a}\\left(\\varepsilon\\_{abc}K^{bf}u^{e}\\nabla\\_{e}K\\_{f}^{c}\\right).\\end{aligned}$$ \nIn the above, *ɛ**a**b**c* is defined in ([eps3def]). The purpose of extracting the total derivatives is to recast the expressions in terms of monomials in the SCG basis. One can also show that\n \n$$\\begin{aligned}\n \\nabla^{c}\\left(\\varepsilon\\_{abc}a^{a}a^{d}K\\_{d}^{b}\\right) & = & \\varepsilon\\_{abc}\\left(-K\\_{e}^{a}a^{b}\\mathrm{D}^{c}a^{e}+a^{a}a^{d}\\mathrm{D}^{c}K\\_{d}^{b}\\right),\\\\\n \\nabla^{c}\\left(\\varepsilon\\_{abc}\\mathrm{D}^{a}a^{d}K\\_{d}^{b}\\right) & = & \\varepsilon\\_{abc}\\left(K\\_{e}^{a}a^{b}\\mathrm{D}^{c}a^{e}-\\,{}^{3}\\!R\\_{e}^{a}K^{be}a^{c}+\\varepsilon\\_{abc}\\mathrm{D}^{a}a^{d}\\mathrm{D}^{c}K\\_{d}^{b}\\right).\\end{aligned}$$ \nParity-violating GST monomials\n==============================\nThe linearly independent GST monomials up to *d* = 4 have been exhausted and classified in (see also ). In this appendix, we list the parity-violating monomials in the cases of *d* = 3 and *d* = 4 for convenience, which are summarized in Table [tab:gstmono].\n[h]\n[tab:gstmono]\nFor the (1; 1, 0) category, there is one monomial\n \n$$F\\_{1}^{(1;1,0)}=\\frac{1}{\\sigma^{3}}\\varepsilon\\_{abcd}R\\_{ef}^{\\phantom{ef} cd}\\phi^{a}\\phi^{e}\\phi^{bf}.$$ \nHere and in what follows, the factor *σ* is a shorthand for $\\sigma\\equiv \\sqrt{-\\phi\\_{a}\\phi^{a}}=\\sqrt{2X}$.\nFor the (0; 2, 1) category, there is one monomial,\n \n$$F\\_{6}^{(0;2,1)}=\\frac{1}{\\sigma^{6}}\\varepsilon\\_{abcd}\\phi^{e}\\phi\\_{f}\\phi^{a}\\phi\\_{e}^{b}\\phi\\_{m}^{c}\\nabla^{m}\\phi^{df}.$$ \nFor the (1; 2, 0) category, there are 10 linearly independent monomials,\n \n$$\\begin{aligned}\n& F\\_{1}^{(1;2,0)}=\\frac{1}{\\sigma^{2}}\\varepsilon\\_{abcd}R\\_{ef}^{\\quad cd}\\phi^{ae}\\phi^{bf}, \\label{app:F1201}\\\\\n& F\\_{2}^{(1;2,0)}=\\frac{1}{\\sigma^{4}}\\varepsilon\\_{abcd}R\\_{ef}^{\\quad cd}\\phi^{a}\\phi^{e}\\phi\\_{m}^{b}\\phi^{fm},\\\\\n & F\\_{3}^{(1;2,0)}=\\frac{1}{\\sigma^{4}}\\varepsilon\\_{abcd}R\\_{ef}^{\\quad cd}\\phi^{e}\\phi^{m}\\phi\\_{m}^{a}\\phi^{fb}, \\\\ & F\\_{4}^{(1;2,0)}=\\frac{1}{\\sigma^{4}}\\varepsilon\\_{abcd}R\\_{ef}^{\\quad cd}\\phi^{a}\\phi^{m}\\phi^{be}\\phi\\_{m}^{f}, \\\\ & F\\_{5}^{(1;2,0)}=\\frac{1}{\\sigma^{4}}\\varepsilon\\_{abcd}R\\_{ef}^{\\quad cm}\\phi^{a}\\phi^{e}\\phi^{bf}\\phi\\_{m}^{d},\\\\ & F\\_{6}^{(1;2,0)}=\\frac{1}{\\sigma^{4}}\\varepsilon\\_{abcd}R^{ae}\\phi^{b}\\phi^{f}\\phi\\_{e}^{c}\\phi\\_{f}^{d},\\\\\n & F\\_{7}^{(1;2,0)}=\\frac{1}{\\sigma^{6}}\\varepsilon\\_{abcd}R\\_{ef}^{\\quad cd}\\phi^{m}\\phi^{n}\\phi^{e}\\phi^{a}\\phi\\_{m}^{f}\\phi\\_{n}^{b},\\quad \\\\\n & F\\_{8}^{(1;2,0)}=\\frac{1}{\\sigma^{6}}\\varepsilon\\_{abcd}R\\_{ef}^{\\quad cm}\\phi^{a}\\phi^{e}\\phi\\_{m}\\phi^{n}\\phi\\_{n}^{b}\\phi^{df},\\\\\n & F\\_{9}^{(1;2,0)}=F\\_{1}^{(1;1,0)}E\\_{1}^{(0;1,0)}=\\frac{1}{\\sigma^{4}}\\varepsilon\\_{abcd}R\\_{ef}^{\\quad cd}\\phi^{a}\\phi^{e}\\phi^{bf}\\phi\\_{m}^{m},\\\\\n & F\\_{10}^{(1;2,0)}=F\\_{1}^{(1;1,0)}E\\_{2}^{(0;1,0)}=\\frac{1}{\\sigma^{6}}\\varepsilon\\_{abcd}R\\_{ef}^{\\quad cd}\\phi^{a}\\phi^{e}\\phi^{bf}\\phi\\_{m}\\phi\\_{n}\\phi^{mn}, \\label{app:F12010}\n \\end{aligned}$$ \nwhere the 2 factorizable monomials *F*9(1; 2, 0) and *F*10(1; 2, 0) are also shown explicitly. For the (2; 0, 0) category, there are 5 linearly independent monomials,\n \n$$\\begin{aligned}\n& F\\_{1}^{(2;0,0)}=\\varepsilon\\_{abcd}R\\_{ef}^{\\quad cd}R^{abef}, \\label{app:F2001}\\\\\n& F\\_{2}^{(2;0,0)}=\\frac{1}{\\sigma^{2}}\\varepsilon\\_{abcd}R\\_{ef}^{\\quad cd}R\\_{\\quad\\;m}^{abf}\\phi^{e}\\phi^{m},\\\\\n & F\\_{3}^{(2;0,0)}=\\frac{1}{\\sigma^{2}}\\varepsilon\\_{abcd}R\\_{ef}^{\\quad cd}R\\_{\\quad\\;m}^{efa}\\phi^{b}\\phi^{m},\\\\\n &F\\_{4}^{(2;0,0)}=\\frac{1}{\\sigma^{2}}\\varepsilon\\_{abcd}R\\_{ef}^{\\quad cd}R^{ae}\\phi^{b}\\phi^{f},\\\\\n & F\\_{5}^{(2;0,0)}=\\frac{1}{\\sigma^{4}}\\varepsilon\\_{abcd}R\\_{ef}^{\\quad cd}R^{amen}\\phi^{b}\\phi^{f}\\phi\\_{m}\\phi\\_{n}. \\label{app:F2005}\\end{aligned}$$ \nFor the (1; 0, 1) category, there is a single independent monomial\n \n$$F\\_{4}^{(1;0,1)}=\\frac{1}{\\sigma^{4}}\\varepsilon\\_{abcd}R\\_{ef}^{\\phantom{ef}cd}\\phi^{a}\\phi^{e}\\phi^{m}\\nabla^{b}\\phi\\_{m}^{f}.$$ \nThe above complete sets of GST monomials derived in are independent in the sense of linear algebra. Some of the monomials are related to each other up to total derivatives. For *F*6(0; 2, 1), after some manipulations, we find\n \n$$F\\_{6}^{(0;2,1)}=-\\frac{1}{2}F\\_{7}^{(1;2,0)}+\\nabla\\_{d}\\left(\\frac{1}{\\sigma^{6}}\\varepsilon\\_{abc}^{\\phantom{abc}d}\\phi^{e}\\phi^{f}\\phi^{a}\\phi\\_{e}^{b}\\phi^{cm}\\phi\\_{mf}\\right). \\label{app:F0216red}$$ \nIt is interesting to note that the total derivative has effectively vanishing contribution if the coefficients are functions of *ϕ* and *X*, since\n \n$$\\begin{aligned}\n & & f\\left(\\phi,X\\right)\\nabla\\_{d}\\left(\\frac{1}{\\sigma^{6}}\\varepsilon\\_{abc}^{\\phantom{abc}d}\\phi^{e}\\phi^{f}\\phi^{a}\\phi\\_{e}^{b}\\phi^{cm}\\phi\\_{mf}\\right) \\\\\n & \\simeq & \\nabla\\_{d}f\\frac{1}{\\sigma^{6}}\\varepsilon\\_{abc}^{\\phantom{abc}d}\\phi^{e}\\phi^{f}\\phi^{a}\\phi\\_{e}^{b}\\phi^{cm}\\phi\\_{mf}\\\\\n & = & \\left(\\frac{\\partial f}{\\partial\\phi}\\phi\\_{d}+\\frac{\\partial f}{\\partial X}\\left(-\\phi\\_{nd}\\phi^{n}\\right)\\right)\\left(\\frac{1}{\\sigma^{6}}\\varepsilon\\_{abc}^{\\phantom{abc}d}\\phi^{e}\\phi^{f}\\phi^{a}\\phi\\_{e}^{b}\\phi^{cm}\\phi\\_{mf}\\right)\\\\\n & \\equiv & 0.\\end{aligned}$$ \nAs a result, ([app:F0216red]) implies that\n \n$$f\\left(\\phi,X\\right)F\\_{6}^{(0;2,1)}\\simeq-\\frac{1}{2}f\\left(\\phi,X\\right)F\\_{7}^{(1;2,0)}.$$ \nFor *F*4(1; 0, 1), we have\n \n$$F\\_{4}^{(1;0,1)}=-F\\_{2}^{(1;2,0)}-F\\_{4}^{(1;2,0)}-4F\\_{7}^{(1;2,0)}+\\nabla^{b}\\left(\\frac{1}{\\sigma^{4}}\\varepsilon\\_{abcd}R\\_{ef}^{\\phantom{ef}cd}\\phi^{a}\\phi^{e}\\phi^{m}\\phi\\_{m}^{f}\\right).$$ \nFor coefficients that are functions of *ϕ* and *X*, we have\n \n$$fF\\_{4}^{(1;0,1)}\\simeq-fF\\_{2}^{(1;2,0)}-fF\\_{4}^{(1;2,0)}-4\\left(f-\\frac{X}{2}\\frac{\\partial f}{\\partial X}\\right)F\\_{7}^{(1;2,0)}. \\label{app:F1014red}$$ \nAs a result, according to Table [tab:gstmono], since *F*6(0; 2, 1) and *F*4(1; 0, 1) can be reduced by integrations by parts, we are left with 15 GST monomials *F*1(1; 2, 0), ⋯, *F*10(1; 2, 0) and *F*1(2; 0, 0), ⋯, *F*5(2; 0, 0). It is also interesting to verify that these 15 GST monomials satisfy 7 identities in the unitary gauge, which we choose to be:\n \n$$\\begin{aligned}\n F\\_{1}^{(2;0,0)}+4F\\_{3}^{(2;0,0)} & \\xlongequal{\\mathrm{u.g.}} & 0, \\label{app:ugid1}\\\\\n F\\_{2}^{(2;0,0)}-4F\\_{5}^{(2;0,0)} & \\xlongequal{\\mathrm{u.g.}} & 0,\\\\\n F\\_{3}^{(1;2,0)}+F\\_{4}^{(1;2,0)}-2F\\_{6}^{(1;2,0)} & \\xlongequal{\\mathrm{u.g.}} & 0,\\\\\n F\\_{1}^{(1;2,0)}-2F\\_{4}^{(1;2,0)}-4F\\_{5}^{(1;2,0)} & \\xlongequal{\\mathrm{u.g.}} & 0,\\\\\n F\\_{3}^{(1;2,0)}-F\\_{7}^{(1;2,0)}+2F\\_{8}^{(1;2,0)}+F\\_{10}^{(1;2,0)} & \\xlongequal{\\mathrm{u.g.}} & 0,\\\\\n F\\_{1}^{(2;0,0)}-8F\\_{4}^{(2;0,0)}+8F\\_{1}^{(1;2,0)}-16F\\_{4}^{(1;2,0)}+32F\\_{6}^{(1;2,0)} & \\xlongequal{\\mathrm{u.g.}} & -16\\mathcal{D},\\\\\n F\\_{2}^{(2;0,0)}+3F\\_{1}^{(1;2,0)}+2F\\_{2}^{(1;2,0)}+6F\\_{3}^{(1;2,0)}-2F\\_{4}^{(1;2,0)}-2F\\_{9}^{(1;2,0)} & \\xlongequal{\\mathrm{u.g.}} & -4\\mathcal{D}, \\label{app:ugid7}\\end{aligned}$$ \nwhere D in the last two identities is the total derivative term defined in ([app:calD]). Therefore, in the sense of the unitary gauge, there are only 8 independent GST monomials of *d* = 4, which is exactly the same number of SCG monomials of the complete basis.\n10\nT.D. Lee and C.-N. Yang, *Question of Parity Conservation in Weak Interactions*, [*Phys. Rev.* **104** (1956) 254](https://doi.org/10.1103/PhysRev.104.254).\nC.S. Wu, E. Ambler, R.W. Hayward, D.D. Hoppes and R.P. Hudson, *Experimental Test of Parity Conservation in *β* Decay*, [*Phys. Rev.* **105** (1957) 1413](https://doi.org/10.1103/PhysRev.105.1413).\nY. Minami and E. Komatsu, *New Extraction of the Cosmic Birefringence from the Planck 2018 Polarization Data*, [*Phys. Rev. Lett.* **125** (2020) 221301](https://doi.org/10.1103/PhysRevLett.125.221301) [[2011.11254](https://arxiv.org/abs/2011.11254)].\nJ.R. Eskilt and E. Komatsu, *Improved constraints on cosmic birefringence from the WMAP and Planck cosmic microwave background polarization data*, [*Phys. Rev. D* **106** (2022) 063503](https://doi.org/10.1103/PhysRevD.106.063503) [[2205.13962](https://arxiv.org/abs/2205.13962)].\nE. Komatsu, *New physics from the polarized light of the cosmic microwave background*, [*Nature Rev. Phys.* **4** (2022) 452](https://doi.org/10.1038/s42254-022-00452-4) [[2202.13919](https://arxiv.org/abs/2202.13919)].\nL. Caloni, S. Giardiello, M. Lembo, M. Gerbino, G. Gubitosi, M. Lattanzi et al., *Probing Lorentz-violating electrodynamics with CMB polarization*, [*JCAP* **03** (2023) 018](https://doi.org/10.1088/1475-7516/2023/03/018) [[2212.04867](https://arxiv.org/abs/2212.04867)].\nN.A. Nilsson and C. Le Poncin-Lafitte, *Reexamining aspects of spacetime-symmetry breaking with CMB polarization*, [*Phys. Rev. D* **109** (2024) 015032](https://doi.org/10.1103/PhysRevD.109.015032) [[2311.16368](https://arxiv.org/abs/2311.16368)].\nR. Hojman, C. Mukku and W.A. Sayed, *PARITY VIOLATION IN METRIC TORSION THEORIES OF GRAVITATION*, [*Phys. Rev. D* **22** (1980) 1915](https://doi.org/10.1103/PhysRevD.22.1915).\nE. Witten, *Some Properties of O(32) Superstrings*, [*Phys. Lett. B* **149** (1984) 351](https://doi.org/10.1016/0370-2693(84)90422-2).\nM.B. Green and J.H. Schwarz, *Anomaly Cancellation in Supersymmetric D=10 Gauge Theory and Superstring Theory*, [*Phys. Lett. B* **149** (1984) 117](https://doi.org/10.1016/0370-2693(84)91565-X).\nN.E. Mavromatos, *CPT violation and decoherence in quantum gravity*, [*Lect. Notes Phys.* **669** (2005) 245](https://doi.org/10.1007/11377306_8) [[gr-qc/0407005](https://arxiv.org/abs/gr-qc/0407005)].\nL. Freidel, D. Minic and T. Takeuchi, *Quantum gravity, torsion, parity violation and all that*, [*Phys. Rev. D* **72** (2005) 104002](https://doi.org/10.1103/PhysRevD.72.104002) [[hep-th/0507253](https://arxiv.org/abs/hep-th/0507253)].\nLIGO Scientific, Virgo collaboration, *Observation of Gravitational Waves from a Binary Black Hole Merger*, [*Phys. Rev. Lett.* **116** (2016) 061102](https://doi.org/10.1103/PhysRevLett.116.061102) [[1602.03837](https://arxiv.org/abs/1602.03837)].\nLIGO Scientific, Virgo collaboration, *GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral*, [*Phys. Rev. Lett.* **119** (2017) 161101](https://doi.org/10.1103/PhysRevLett.119.161101) [[1710.05832](https://arxiv.org/abs/1710.05832)].\nLISA collaboration, *Laser Interferometer Space Antenna*, [1702.00786](https://arxiv.org/abs/1702.00786).\nS. Kawamura et al., *The Japanese space gravitational wave antenna: DECIGO*, [*Class. Quant. Grav.* **28** (2011) 094011](https://doi.org/10.1088/0264-9381/28/9/094011).\nB. Sathyaprakash et al., *Scientific Objectives of Einstein Telescope*, [*Class. Quant. Grav.* **29** (2012) 124013](https://doi.org/10.1088/0264-9381/29/12/124013,\n 10.1088/0264-9381/30/7/079501) [[1206.0331](https://arxiv.org/abs/1206.0331)].\nW.-R. Hu and Y.-L. Wu, *The Taiji Program in Space for gravitational wave physics and the nature of gravity*, [*Natl. Sci. Rev.* **4** (2017) 685](https://doi.org/10.1093/nsr/nwx116).\nW.-H. Ruan, Z.-K. Guo, R.-G. Cai and Y.-Z. Zhang, *Taiji program: Gravitational-wave sources*, [*Int. J. Mod. Phys. A* **35** (2020) 2050075](https://doi.org/10.1142/S0217751X2050075X) [[1807.09495](https://arxiv.org/abs/1807.09495)].\nTianQin collaboration, *TianQin: a space-borne gravitational wave detector*, [*Class. Quant. Grav.* **33** (2016) 035010](https://doi.org/10.1088/0264-9381/33/3/035010) [[1512.02076](https://arxiv.org/abs/1512.02076)].\nH. Li et al., *Probing Primordial Gravitational Waves: Ali CMB Polarization Telescope*, [*Natl. Sci. Rev.* **6** (2019) 145](https://doi.org/10.1093/nsr/nwy019) [[1710.03047](https://arxiv.org/abs/1710.03047)].\nS.H. Alexander and N. Yunes, *Gravitational wave probes of parity violation in compact binary coalescences*, [*Phys. Rev.* **D97** (2018) 064033](https://doi.org/10.1103/PhysRevD.97.064033) [[1712.01853](https://arxiv.org/abs/1712.01853)].\nW. Zhao, T. Zhu, J. Qiao and A. Wang, *Waveform of gravitational waves in the general parity-violating gravities*, [*Phys. Rev.* **D101** (2020) 024002](https://doi.org/10.1103/PhysRevD.101.024002) [[1909.10887](https://arxiv.org/abs/1909.10887)].\nM. Satoh, S. Kanno and J. Soda, *Circular Polarization of Primordial Gravitational Waves in String-inspired Inflationary Cosmology*, [*Phys. Rev.* **D77** (2008) 023526](https://doi.org/10.1103/PhysRevD.77.023526) [[0706.3585](https://arxiv.org/abs/0706.3585)].\nM. Satoh, *Slow-roll Inflation with the Gauss-Bonnet and Chern-Simons Corrections*, [*JCAP* **11** (2010) 024](https://doi.org/10.1088/1475-7516/2010/11/024) [[1008.2724](https://arxiv.org/abs/1008.2724)].\nP. Creminelli, J. Gleyzes, J. Noreña and F. Vernizzi, *Resilience of the standard predictions for primordial tensor modes*, [*Phys. Rev. Lett.* **113** (2014) 231301](https://doi.org/10.1103/PhysRevLett.113.231301) [[1407.8439](https://arxiv.org/abs/1407.8439)].\nD. Cannone, J.-O. Gong and G. Tasinato, *Breaking discrete symmetries in the effective field theory of inflation*, [*JCAP* **1508** (2015) 003](https://doi.org/10.1088/1475-7516/2015/08/003) [[1505.05773](https://arxiv.org/abs/1505.05773)].\nA. Lue, L.-M. Wang and M. Kamionkowski, *Cosmological signature of new parity violating interactions*, [*Phys. Rev. Lett.* **83** (1999) 1506](https://doi.org/10.1103/PhysRevLett.83.1506) [[astro-ph/9812088](https://arxiv.org/abs/astro-ph/9812088)].\nS. Saito, K. Ichiki and A. Taruya, *Probing polarization states of primordial gravitational waves with CMB anisotropies*, [*JCAP* **09** (2007) 002](https://doi.org/10.1088/1475-7516/2007/09/002) [[0705.3701](https://arxiv.org/abs/0705.3701)].\nC.R. Contaldi, J. Magueijo and L. Smolin, *Anomalous CMB polarization and gravitational chirality*, [*Phys. Rev. Lett.* **101** (2008) 141101](https://doi.org/10.1103/PhysRevLett.101.141101) [["},"data_id":{"kind":"string","value":"arxiv_0185560"}}},{"rowIdx":46395,"cells":{"content":{"kind":"string","value":"varepsilon}\\_h}]\\_E/\\partial s \\, \\nu\\_E)$ and its extension *P**E**ρ**E* define the test function *ϕ* : = *b**P**E**ρ**E* ∈ *C*0∞(*ω**E*) in Lemma [lemma:representation]. The representation formula and (*ɛ*(*v*), Curl2*ϕ*)*L*2(*ω**E*) = 0 lead to \n$$\\| b^{1/2} \\rho\\_E\\|\\_{L^2(E)}^2\n= (\\varepsilon(v)-{\\bm{\\varepsilon}\\_h},{\\operatorname{Curl}}^2\\phi)\\_{L^2(\\omega\\_E)} + ({\\operatorname{rot}}\\_{NC}^2{\\bm{\\varepsilon}\\_h},\\phi)\\_{L^2(\\omega\\_E)}.$$ \n The inverse inequality ∥*ρ**E*∥*L*2(*E*)∥ ≤ ∥*b*1/2*ρ**E*∥*L*2(*E*), Cauchy inequalities, and the right scaling properties of *ϕ* lead to \n$$\\|\\rho\\_E\\|\\_{L^2(E)}^2\\lesssim \\|\\rho\\_E\\|\\_{L^2(E)}\\left( \nh\\_E^{-3/2} \\|{\\bm{\\varepsilon}\\_h}-\\varepsilon(v) \\|\\_{L^2(\\omega\\_E)} + h\\_E^{1/2}\\| {\\operatorname{rot}}\\_{NC}^2{\\bm{\\varepsilon}\\_h}\\| \\_{L^2(\\omega\\_E)} \\right).$$ \n This and Lemma [lemma:eff:1] conclude the proof.\nThe efficiency of the last edge contribution involves the second Dirichlet data oscillation osc*I**I*(*u**D*, *E*) from.\n[efficiency of second boundary edge residual][lemma:eff:5] Any function *v* ∈ *H*1(*ω**E*; R2) with *v*∣*E* = *u**D*∣*E* along *E* ∈ E(Γ*D*) satisfies \n$$h\\_E^{3/2} \\|\\tau\\_E\\cdot {\\operatorname{rot}}{\\bm{\\varepsilon}\\_h}- \\nu\\_E\\cdot (\\frac{ \\partial {\\bm{\\varepsilon}\\_h}\\tau\\_E }{\\partial s }\n+\\frac{ \\partial^2 u\\_D}{ \\partial s^2}) \\|\\_{L^2(E)}\n\\lesssim \\| {\\bm{\\varepsilon}\\_h}- \\varepsilon(v) \\|\\_{L^2(\\omega\\_E)} + {\\operatorname{osc}}\\_{II}(u\\_D,E).$$ \nSelect a maximal open ball *B*(*x**E*, 2*r**E*) ∩ Ω ⊂ *ω**E* around a point *x**E* ∈ *E* with maximal radius 2*r**E* such that *B*(*x**E*, 2*r**E*) ∩ *ω**E* is a half ball. The regularization *b* : = *χ**B*(*x**E*, *r**E*) \\* *η**r**E* ∈ *C**c*∞(R2) of the characteristic function *χ**B*(*x**E*, *r**E*) attains values in [0, 1] and a positive integral mean *h**E*− 1∫*E**b* *d**s* ≈ 1 along *E* (depending only on the shape regularity of T); *b* vanishes on ∂*ω**E* \\ *E* and its normal derivative ∇*b* ⋅ *ν* = 0 vanishes along the entire boundary ∂*ω**E*.\nThe Pythagoras theorem ∥*ρ*∥*L*2(*E*)2 = ∥*ρ**E*∥*L*2(*E*)2 + *h**E*− 3osc*I**I*2(*u**D*, *E*) for the residual $\\rho:=\\tau\\_E\\cdot {\\operatorname{rot}}{\\bm{\\varepsilon}\\_h}- \\nu\\_E\\cdot (\\frac{ \\partial {\\bm{\\varepsilon}\\_h}\\tau\\_E }{\\partial s }\n +\\frac{ \\partial^2 u\\_D}{ \\partial s^2})$ and its *L*2 projection *ρ**E* : = Π*m*, *E**ρ* onto *P**m*(*E*) reduces the proof to the estimation of ∥*ρ**E*∥*L*2(*E*). The normal derivative of $\\phi:= b\\, P\\_E \\rho\\_E\\in C^\\infty(\\overline{\\omega\\_E})$ vanishes along the boundary ∂*ω**E* and Lemma [lemma:representation] shows \n$$({\\operatorname{rot}}{\\bm{\\varepsilon}\\_h}-\\frac{ \\partial {\\bm{\\varepsilon}\\_h}\\nu\\_E}{\\partial s}, b \\rho\\_E \\tau\\_E)\\_{L^2(E)}\n= ({\\operatorname{rot}}\\_{NC}^2{\\bm{\\varepsilon}\\_h},\\phi)\\_{L^2(\\omega\\_E)}-({\\bm{\\varepsilon}\\_h}, {\\operatorname{Curl}}^2 \\phi)\\_{L^2(\\omega\\_E)}.$$ \n The arguments in Lemma [lemma:bd] show (∂2*u**D*/∂*s*2, *b**ρ**E* *ν**E*)*L*2(*E*) = (*ɛ*(*v*), Curl2*ϕ*)*L*2(*ω**E*). The combination of the two identities leads to a formula for (*ρ*, *b**ρ**E*)*L*2(*E*). Since *ρ* − *ρ**E* is controlled in osc*I**I*2(*u**D*, *E*), this and an inverse inequality in the beginning result in\n \n$$\\begin{aligned}\n&\\|\\rho\\_E\\|\\_{L^2(E)}^2\\lesssim (b\\rho\\_E,\\rho\\_E) \\_{L^2(E)}\n=(\\rho, b\\rho\\_E) \\_{L^2(E)}- (\\rho-\\rho\\_E, b\\rho\\_E) \\_{L^2(E)} \\\\\n&\\; \\lesssim ({\\operatorname{rot}}\\_{NC}^2{\\bm{\\varepsilon}\\_h},\\phi)\\_{L^2(\\omega\\_E)}\n-({\\bm{\\varepsilon}\\_h}-\\varepsilon(v), {\\operatorname{Curl}}^2 \\phi)\\_{L^2(\\omega\\_E)} +\\|\\rho\\_E\\|\\_{L^2(E) }h\\_E^{-3/2} {\\operatorname{osc}}\\_{II}(u\\_D,E).\\end{aligned}$$ \nThe scaling properties of *ϕ* and its derivatives are as in the proof of the previous lemma. With Lemma [lemma:eff:1] in the end, this concludes the proof.\nNumerical examples\n==================\nThis section is devoted to numerical experiments for four different domains to demonstrate robustness in the reliability and efficiency of the a posteriori error estimator *η*(Tℓ, *σ*ℓ). The implementation follows for *k* = 1 with Lamé parameters *λ* and *μ* from *λ* = *E**ν*/((1 + *ν*)(1 − 2*ν*)) and *μ* = *E*/(2(1 + *ν*)) for a Young’s modulus *E* = 105 and various Poisson ratios *ν* = 0.3 and *ν* = 0.4999.\nAcademic example\n----------------\n\n[fig:Convergence1]\nThe model problem on the unit square Ω = (0, 1)2 with homogeneous Dirichlet boundary conditions and the right-hand side *f* = (*f*1, *f*2), \n*f*1(*x*, *y*) = − *f*2(*y*, *x*) = − 2*μ**π*3cos(*π**y*)sin(*π**y*)(2cos(2*π**x*) − 1) for (*x*, *y*) ∈ Ω, \n allows the smooth exact solution \n*u*(*x*, *y*) = *π* sin(*π**x*)sin(*π**y*) (cos(*π**y*)sin(*π**x*), − cos(*π**x*)sin(*π**y*)) for (*x*, *y*) ∈ Ω. \n Note that *f* depends only on the Lamé parameter *μ* and not on the critical Lamé parameter *λ*. For uniform mesh refinement, Figure [fig:Convergence1] displays the robust third-order convergence of the a posteriori error estimator *η*(Tℓ, *σ*ℓ) as well as the Arnold-Winther finite element stress error. The convergence is robust in the Poisson ratio *ν* → 1/2 and the a posteriori error estimator proves to be reliable and efficient. In this example, the oscillations osc(*f*, Tℓ) dominate the a posteriori error estimator.\nThis typical observation motivates numerical examples with *f* ≡ 0 in the sequel.\nCircular inclusion\n------------------\n[scale = 4]\n(0.25,0) – (1,0); (1,0) – (1,1); (1,1) – (0,1); (0,0.25) – (0,1);\n(0.25,0) arc (0:90:0.25);\nin 0,...,9 (1,0.1+0.05) – ++(0.1,0); \nin 3,...,9 (-0.05,0.1+0.05) circle (0.035); \nin 3,...,9 (0.1+0.05,-0.05) circle (0.035); \n(0.1,0.1) node Γ*D*;\n(1.2,0.5) node Γ*N*;\n(-0.15,0) – (-0.15,0.25); (-0.2,0.1) node[rotate=90] 25;\n(-0.15,0.25) – (-0.15,1); (-0.2,0.6) node[rotate=90] 75;\n(0,-0.15) – (0.25,-0.15); (0.1,-0.2) node 25;\n(0.25,-0.15) – (1,-0.15); (0.6,-0.2) node 75;\nin 1,...,9 (9:0.25) – (9:0.2); \n[fig:ex:geom:CircularIncusion]\n\n[fig:Convergence2]\nThe second benchmark example from the literature models a rigid circular inclusion in an infinite plate for the domain Ω with rather mixed boundary conditions indicated with mechanical symbols in Figure [fig:ex:geom:CircularIncusion]. The exact solution to the model problem reads (with polar coordinates (*r*, *ϕ*) and *κ* = 3 − 4*ν*, *γ* = 2*ν* − 1, *a* = 1/4)\n \n$$\\begin{aligned}\n u\\_r &= \\frac{1}{8\\mu r}\\left( (\\kappa-1)r^2 + 2\\gamma a^2 + \n \\left( 2r^2 -\\frac{2(\\kappa+1)a^2}{\\kappa} + \\frac{2 a^4}{\\kappa r^2}\\right) \\cos(2\\phi)\\right),\\\\\n u\\_\\phi &= -\\frac{1}{8\\mu r}\\left( 2r^2-\\frac{2(\\kappa-1)a^2}{\\kappa} - \\frac{2a^4}{\\kappa r^2}\\right)\\sin(2\\phi).\\end{aligned}$$ \nThe approximation of the circular inclusion through a polygon is rather critical for the higher-order Arnold–Winther MFEM. In the absence of an implementation of parametric boundaries, adaptive mesh refinement is necessary for higher improvements. The adaptive algorithm of this section is the same for all examples and acts on polygons; in particular, it does not monitor the curved boundary, but whenever some edge at the curved part Γ*D* is refined in this example, the midpoint is a new node and projected onto Γ*D*. The convergence history plot in Figure [fig:Convergence2] shows a reduced convergence for uniform refinement, while adaptive refinement (of the circular boundary) leads to optimal third-order convergence.\nL-shaped benchmark\n------------------\n[scale=1.25] (0,0) coordinate (A); (-1,-1) coordinate (B); (0,-2) coordinate (C); (2,0) coordinate (D); (0,2) coordinate (E); (-1,1) coordinate (F);\n(A) – (B) – (C) – (D) – (E) – (F) – (A);\n(0,0) – (2.5,0); (0,0) – (0,2.5);\n(2.5,-0.1) node *x*; (-0.1,2.5) node *y*; (1.3,1.3) node Γ*D*;\n(0,0) – (0.5,0.5); (0.25,0.5) node *r*;\n(0.5,0) arc (0:45:0.5); (0.55,0.25) node *ϕ*;\n(-1,-2.25) – (0,-2.25); (0,-2.25) – (2,-2.25);\n(-1.25,-2) – (-1.25,2); (-0.5,-2.4 ) node 1; (1,-2.4) node 2; (-1.4,0) node[rotate=90] 4;\n(-0.5,0) node Γ*N*;\nin 0,...,10 (-1+0.1,-1-0.1) – ++(0,-0.15); (-1+0.1,1+0.1) – ++(0,0.15); \nin 0,...,20 (0.1,-2+0.1) – ++(0,-0.15); (0.1,2-0.1) – ++(0,0.15); \n[fig:geom:Lshape]\n\n[fig:Convergence3]\nConsider the rotated L-shaped domain with Dirichlet and Neumann boundary depicted in Figure [fig:geom:Lshape]. The exact solution reads in polar coordinates\n \n$$\\begin{aligned}\n u\\_r(r,\\phi) &= \\frac{r^{\\alpha}}{2\\mu} \\left( -(\\alpha+1)\\cos((\\alpha+1)\\phi) + (C\\_2-\\alpha-1)C\\_1\\cos((\\alpha-1)\\phi)\\right),\\\\\n u\\_\\phi(r,\\phi) &= \\frac{r^{\\alpha}}{2\\mu} \\left( (\\alpha+1)\\sin((\\alpha+1)\\phi) + (C\\_2+\\alpha-1)C\\_1\\sin((\\alpha-1)\\phi)\\right).\\end{aligned}$$ \nThe constants are *C*1 : = − cos((*α* + 1)*ω*)/cos((*α* − 1)*ω*) and *C*2 : = 2(*λ* + 2*μ*)/(*λ* + *μ*), where *α* = 0.544483736782 is the first root of *α*sin(2*ω*) + sin(2*ω**α*) = 0 for *ω* = 3*π*/4. The volume force *f* ≡ 0 and the Neumann boundary data *g* ≡ 0 vanish, and the Dirichlet boundary conditions *u**D* are extracted from the exact solution.\nFigure [fig:Convergence3] shows suboptimal convergence O(*N*ℓ− 0.27), namely an expected rate *α* in terms of the maximal mesh-size, for uniform and fourth-order *L*2 stress convergence for adaptive mesh-refinement.\nDespite the singular solution, the adaptive algorithm recovers the higher convergence of Theorem [supperconvergence] as in.\nCook membrane problem\n---------------------\n[scale = 0.08]\n(0,0) coordinate (A); (0,44) coordinate (B); (48,60) coordinate (C); (48,44) coordinate (D);\n(D) – (A) – (B) – (C); (B) – (D);\nin 1,...,21 (0,2) – ++(-3,-3); (48,44) – (48,52); (48,52) – (48,60); (48,60) – (48,65);\n(-6,22) node[fill=white] Γ*D* ; (54,53) node Γ*N*;\n(0,-5) – (48,-5); (24,-8) node 48;\n(-10,0) – (-10,44); (-10,44) – (-10,60); (-15,22) node[rotate=90] 44; (-15,52) node[rotate=90] 16;\n(A) node[right] A; (B) node[left] D; (C) node[right] C; (D) node[below] B;\n[fig:geom:Cooks]\n\n[fig:Convergence4]\nOne of the more popular benchmarks in computational mechanics is the tapered panel Ω with the vertices *A*, *B*, *C*, *D* of Figure [fig:geom:Cooks] clamped on the left side Γ*D* = conv(*D*, *A*) (with *u**D* ≡ 0) under no volume force (*f* ≡ 0) but applied surface tractions *g* = (0, 1) along conv(*B*, *C*) and traction free on the remaining parts conv(*A*, *B*) and conv(*C*, *D*) along the Neumann boundary.\nThis example is a particular difficult one for the Arnold–Winther MFEM because of the incompatible Neumann boundary conditions on the right corners. That means, although *g* is piecewise constant, *g* does not belong to *G*(T) for any triangulation. In the two Neumann corner vertices *B* and *C* we therefore strongly impose the values *σ*ℓ(*B*) = (0.2491, 0.7283; 0.7283, 0.6676) and *σ*ℓ(*C*) = (3/20, 11/20; 11/20, 11/60) for the design of *g*ℓ ∈ *G*(Tℓ).\nSince the exact solution is unknown, the error approximation rests on a reference solution $\\tilde \\sigma$ computed as *P*5(T) displacement approximation on the uniform refinement of the finest adapted triangulation.\nThe large pre-asymptotic range of the convergence history plot in Figure [fig:Convergence4] illustrates the difficulties of the Arnold-Winther finite element method in case of incompatible Neumann boundary conditions according to its nodal degrees of freedom. Once the resulting and dominating boundary oscillations (caused by the necessary choice of discrete compatible Neumann conditions in *G*(Tℓ)) osc(*g* − *g*ℓ, Eℓ(Γ*N*)) are resolved through adaptive mesh-refining, even the fourth-order *L*2 stress convergence is visible in a long asymptotic range in (the approximated error and) the equivalent error estimator.\nThis example underlines that adaptive mesh-refining is unavoidable in computational mechanics with optimal rates and a large saving in computational time and memory compared to naive uniform mesh-refining.\nAcknowledgements\n================\nThe work has been written, while the three authors enjoyed the hospitality of the Hausdorff Research Institute of Mathematics in Bonn, Germany, during the Hausdorff Trimester Program *Multiscale Problems: Algorithms, Numerical Analysis and Computation*. The research of the first author (CC) has been supported by the Deutsche Forschungsgemeinschaft in the Priority Program 1748 ‘*Reliable simulation techniques in solid mechanics. Development of non-standard discretization methods, mechanical and mathematical analysis*’ under the project ‘*Foundation and application of generalized mixed FEM towards nonlinear problems in solid mechanics*’ (CA 151/22-1). The second author (DG) has been supported by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173. The third author (JG) has been funded by the Austrian Science Fund (FWF) through the project P 29197-N32.\n0\nD. N. Arnold, F. Brezzi, and J. Douglas. : A new mixed finite element for plane elasticity., 1:347–367, 1984.\nD. N. Arnold, R. S. Falk, and R. Winther. Finite element exterior calculus, homological techniques, and applications., 15:1–155, 2006.\nA. Alonso. Error estimators for a mixed method., 74(4):385–395, 1996.\nD. N. Arnold. Differential complexes and numerical stability. In *Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002)*, pages 137–157. Higher Ed. Press, Beijing, 2002.\nD. N. Arnold and R. Winther. Mixed finite elements for elasticity., 92(3):401–419, 2002.\nD. Boffi, F. Brezzi, and M. Fortin., volume 44 of *Springer Series in Computational Mathematics*. Springer, Heidelberg, 2013.\nD. Braess.. Cambridge University Press, Cambridge, third edition, 2007.\nS. C. Brenner and L. R. Scott., volume 15 of *Texts in Applied Mathematics*. Springer, New York, third edition, 2008.\nC. Carstensen. A posteriori error estimate for the mixed finite element method., 66(218):465–476, 1997.\nC. Carstensen. A unifying theory of a posteriori finite element error control., 100(4):617–637, 2005.\nC. Carstensen and G. Dolzmann. A posteriori error estimates for mixed FEM in elasticity., 81(2):187–209, 1998.\nC. Carstensen, M. Eigel, and J. Gedicke. Computational competition of symmetric mixed FEM in linear elasticity., 200(41-44):2903–2915, 2011.\nC. Carstensen, M. Feischl, M. Page, and D. Praetorius. Axioms of adaptivity., 67(6):1195–1253, 2014.\nC. Carstensen and J. Gedicke. Robust residual-based a posteriori Arnold-Winther mixed finite element analysis in elasticity., 300:245–264, 2016.\nC. Carstensen, D. Günther, J. Reininghaus, and J. Thiele. The Arnold-Winther mixed FEM in linear elasticity. Part I: Implementation and numerical verification., 197:3014–3023, 2008.\nC. Carstensen, D. Gallistl, and M. Schedensack. best approximation of the elastic stress in the Arnold-Winther FEM., 36(3):1096–1119, 2016.\nC. Carstensen and J. Hu. A unifying theory of a posteriori error control for nonconforming finite element methods., 100(3):473–502, 2007.\nP. G. Ciarlet., volume 4 of *Studies in Mathematics and its Applications*. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978.\nC. Carstensen, D. Peterseim, and A. Schröder. The norm of a discretized gradient in *H*(div)\\* for a posteriori finite element error analysis., 132:519–539, 2016.\nC. Carstensen and H. Rabus. Axioms of adaptivity for separate marking., arXiv:1606.02165, 2016.\nV. Girault and L. R. Scott. Hermite interpolation of nonsmooth functions preserving boundary conditions., 71(239):1043–1074, 2002.\nJ. Huang, X. Huang, and Y. Xu. Convergence of an adaptive mixed finite element method for Kirchhoff plate bending problems., 49(2):574–607, 2011.\nR. Kouhia and R. Stenberg. A linear nonconforming finite element method for nearly incompressible elasticity and Stokes flow., 124(3):195–212, 1995.\nR. Verfürth.. Wiley and Teubner, 1996.\nAppendix: Fourth-order convergence of the stress in *L*2\n========================================================\nThis appendix explains a high-order convergence phenomenon observed in some numerical benchmark examples for the lowest-order Arnold-Winther method. Adopt the notation from this paper for *k* = 1 and let *σ* solve and let *σ**h* ∈ *A**W**k*(T) solve.\n[fourth-order convergence][supperconvergence] Suppose *f* = *f**h* ∈ *P*1(T; R2) and *g* = *g**h* ∈ *G*(T) and suppose that the stress *σ* ∈ *H*4(Ω; S). Then the *L*2 stress error satisfies (with the maximal mesh-size *h*) \n$$\\|\\sigma-\\sigma\\_h\\|\\_{L^2(\\Omega)} \\lesssim h^4 \\|\\sigma\\|\\_{H^4(\\Omega)}.$$ \nSince the stress error *σ* − *σ**h* ∈ *H*4(T; S) is divergence-free, *α* vanishes in and *σ* − *σ**h* = Curl2*β* ∈ *H*4(T; S). Since *β* ∈ *H*2(T) is piecewise in *C*2, it follows $\\beta\\in C^1(\\overline{\\Omega})$. The Arnold-Winther finite elements have nodal degrees of freedom at the vertices and hence *σ**h* is continuous at each vertex *z* ∈ N. Hence the second derivatives of $\\beta\\in C^2({\\mathcal{T}})\\cap C^1(\\overline{\\Omega})$ are continuous at each vertex *z* ∈ N. It follows that the nodal interpolation operator *I**A* associated to the Argyris finite element space *A*(T) ⊂ *C*1(Ω) ∩ *P*5(T) exists for *β* in the classical sense and is composed of the piecewise local interpolation. This defines *β**h* = *I**A**β* and the divergence-free *τ**h* : = Curl2*β**h* ∈ *A**W*(T) test function in and in. Consequently, \n∥*σ* − *σ**h*∥*L*2(Ω)2 = (*σ* − *σ**h*, Curl2(*β* − *β**h*))*L*2(Ω) ≤ ∥*σ* − *σ**h*∥*L*2(Ω) ∣*β* − *I**A**β*∣*H*2(Ω). \n This and standard local interpolation error estimates for the nodal interpolation of the quintic Argyris finite elements show \n$$\\|\\sigma-\\sigma\\_h\\|\\_{L^2(\\Omega)}\\lesssim h^4 |\\beta |\\_{H^6({\\mathcal{T}})}=h^4 |\\sigma |\\_{H^4(\\Omega)}.$$ \n (With *σ* − *σ**h* = Curl2*β* and ∣*σ**h*∣*H*4(T) = 0 for piecewise cubic *σ**h* in the last step.)\n---\n1. Humboldt-Universität zu Berlin, Berlin, Germany[↩](#fnref1)\n2. Karlsruher Institut für Technologie, Karlsruhe, Germany[↩](#fnref2)\n3. \n\nResidual-based a posteriori error analysis for symmetric mixed Arnold-Winther FEM\n=================================================================================\nThis paper introduces an explicit residual-based a posteriori error analysis for the symmetric mixed finite element method in linear elasticity after Arnold-Winther with pointwise symmetric and *H*(div)-conforming stress approximation. Opposed to a previous publication, the residual-based a posteriori error estimator of this paper is reliable and efficient and truly explicit in that it solely depends on the symmetric stress and does neither need any additional information of some skew symmetric part of the gradient nor any efficient approximation thereof. Hence it is straightforward to implement an adaptive mesh-refining algorithm obligatory in practical computations.\nNumerical experiments verify the proven reliability and efficiency of the new a posteriori error estimator and illustrate the improved convergence rate in comparison to uniform mesh-refining. A higher convergence rates for piecewise affine data is observed in the *L*2 stress error and reproduced in non-smooth situations by the adaptive mesh-refining strategy.\n**Keywords** linear elasticity, mixed finite element method, a posteriori, explicit residual-based, error estimator, reliability, efficiency, symmetric stress finite elements, Arnold-Winther finite element\n**AMS subject classification** 65N15, 65N30\nIntroduction\n============\nOverview\n--------\nThe design of a pointwise symmetric stress approximation $\\sigma\\_h\\in L^2(\\Omega;\\mathbb S)$ with divergence in *L*2(Ω; R*d*), written $\\sigma\\_h \\in H({\\operatorname{div}},\\Omega; \\mathbb S)$, has been a long-standing challenge and the first positive examples in initiated what nowadays is called the finite element exterior calculus. The a posteriori error analysis of mixed finite element methods in elasticity started with on PEERS, where the asymmetric stress approximation *γ**h* arises in the discretization as a Lagrange multiplier to enforce weakly the stress symmetry. This allows the treatment of the term C− 1*σ**h* + *γ**h* as an approximation of the (non symmetric) functional matrix *D**u* for the displacement field with the arguments of developed for mixed finite element schemes for a Poisson model problem. Here and throughout, C denotes a fourth-order elasticity tensor with two Lamé constants *λ* and *μ* and C− 1 is its inverse. Mixed finite element methods appear attractive in the incompressible limit for they typically avoid the locking phenomenon as *λ* → ∞.\nFor mixed finite element methods like the symmetric Arnold-Winther finite element schemes, the subtle term is the nonconforming residual: Given any piecewise polynomial $\\sigma\\_h \\in H({\\operatorname{div}},\\Omega; \\mathbb S)$, compute an upper bound *η*(T, *σ**h*) of \n$$\\inf \\_{v\\in V} \\| {\\mathbb{C}}^{-1/2} \\sigma\\_h-{\\mathbb{C}}^{1/2}\\varepsilon (v) \\|\\_{L^2(\\Omega)} \\lesssim \\eta(\\mathcal{T},\\sigma\\_h).$$ \n Despite general results in this direction, this task had been addressed only by the computation of an approximation to the optimal *v* with Green strain *ε*(*v*) : = sym*D**v* or of some skew-symmetric approximation *γ**h* motivated from the first results in on PEERS. In fact, *any* choice of a piecewise smooth and pointwise skew-symmetric *γ**h* allows for an a posteriori error control of the symmetric stress error *σ* − *σ**h* in. Its efficiency, however, depends on the (unknown and uncontrolled) efficiency of the choice of *γ**h* as an approximation to the skew-symmetric part *γ* of *D**u*.\nThis paper proposes the first reliable and efficient *explicit* residual-based a posteriori error estimator of the nonconforming residual with the typical contributions to *η*(T, *σ**h*) computed from the (known) Green strain approximation ${\\bm{\\varepsilon}\\_h}:= {\\mathbb{C}}^{-1} \\sigma\\_h$. Besides oscillations of the applied forces in the volume and along the Neumann boundary, there is a volume contribution $h\\_T^2\\| {\\operatorname{rot}}{\\operatorname{rot}}{\\bm{\\varepsilon}\\_h}\\|\\_{L^2(T)}$ for each triangle *T* ∈ T and an edge contribution with the jump $[{\\bm{\\varepsilon}\\_h}]\\_E$ across an interior edge *E* with unit normal *ν**E*, tangential unit vector *τ**E*, and length *h**E*, namely \n$$h\\_E^{1/2} \\| \\tau\\_E \\cdot [{\\bm{\\varepsilon}\\_h}]\\_E \\tau\\_E \\|\\_{L^2(E)} +\n h\\_E^{3/2} \\| \\tau\\_E \\cdot [{\\operatorname{rot}}\\_{NC} {\\bm{\\varepsilon}\\_h}]\\_E - \\partial ( \\nu\\_E \\cdot [{\\bm{\\varepsilon}\\_h}]\\_E \\tau\\_E)/\\partial s \\|\\_{L^2(E)},$$ \n and corresponding modification on the edges on the Dirichlet boundary with the (possibly inhomogeneous) Dirichlet data; cf. Remark [remarkonjumptermvanishes] for some partial simplification of the last term displayed.\nFor the ease of this presentation, the analysis involves explicit calculations in two dimensions without any reference to the exterior calculus but with inhomogeneous Dirichlet and Neumann boundary data. The main result is reliability and efficiency to control the stress error robustly in the sense that the multiplicative generic constants hidden in the notation $\\lesssim$ do neither depend on the (local or global) mesh-size nor on the parameter *λ* > 0 but may depend on *μ* > 0 and on the shape regularity of the underlying triangulation T of the domain Ω into triangles through a lower bound of the minimal angle therein.\nLinear elastic model problem\n----------------------------\nThe elastic body Ω is a simply-connected bounded Lipschitz domain Ω ⊂ R2 in the plane with a (connected) polygonal boundary ∂Ω = Γ*D* ∪ Γ*N* split into parts. The displacement boundary Γ*D* is compact and of positive surface measure, while the traction boundary is the relative open complement Γ*N* = ∂Ω ∖ Γ*D* with outer unit normal vector *ν*. Given *u**D* ∈ *H*1(Ω; R2), the volume force *f* ∈ *L*2(Ω; R2), and the applied surface traction *g* ∈ *L*2(Γ*N*; R2), the linear elastic problem seeks a displacement *u* ∈ *H*1(Ω; R2) and a symmetric stress tensor *σ* ∈ *H*(div, Ω; S) with\n \n$$\\begin{aligned}\n\\label{eq:lame}\n\\begin{split}\n -{\\operatorname{div}}\\sigma = f \\quad\\text{and}\\quad \\sigma = {\\mathbb{C}}\\varepsilon(u)\\quad\\mbox{in } \\Omega,\\\\\n u = u\\_D\\quad\\mbox{on }\\Gamma\\_D, \\qquad \\sigma\\nu = g\\quad\\mbox{ on }\\Gamma\\_N.\n\\end{split} \\end{aligned}$$ \nThroughout this paper, given the Lamé parameters *λ*, *μ* > 0 for isotropic linear elasticity, the positive definite fourth-order elasticity tensor C acts as C*E* : = 2*μ* *E* + *λ* tr(*E*) 12 × 2 on any matrix *E* ∈ S with trace tr(*E*) and the 2 × 2 unit matrix 12 × 2. Note that *u**D* acts in only on Γ*D* and is an extension of the continuous function *u**D* ∈ *C*(Γ*D*; R2) also supposed to belong to the edgewise second order Sobolev space *H*2(E(Γ*D*)) below to allow second derivatives with respect to the arc length along boundary edges.\nMore essential will be a discussion on the precise conditions on the Neumann data *g* and its discrete approximation *g**h* below for they belong to the essential boundary conditions in the mixed finite element method based on the dual formulation.\nIn addition to the set of homogeneous displacements *V* and the aforementioned stress space *H*(div, Ω; S), namely, \n$$V:=\\{v\\in H^1(\\Omega;\\mathbb{R}^2)\\ \\big\\vert\\ v|\\_{\\Gamma\\_D}=0\\}\n\\text{ and }\nH({\\operatorname{div}},\\Omega;\\mathbb{S}) := \\{ \\tau \\in L^2(\\Omega;\\mathbb{S}) \\ \\big\\vert\\ {\\operatorname{div}}\\tau\\in L^2(\\Omega;\\mathbb{R}^2)\\},$$ \n and with the exterior unit normal vector *ν* along ∂Ω, the inhomogeneous stress space \n$$\\Sigma(g) := \\Big\\{ \\sigma\\in H({\\operatorname{div}},\\Omega;\\mathbb{S})\\ \\big\\vert \n \\int\\_{\\Gamma\\_N}\\psi\\cdot(\\sigma\\nu )\\;ds = \\int\\_{\\Gamma\\_N}\\psi\\cdot g\\;ds \\text{ for all } \\psi\\in V \\Big\\}$$ \n is defined with respect to the Neumann data *g* ∈ *L*2(Γ*N*) and, in particular, Σ0 : = Σ(0) abbreviates the stress space with homogeneous Neumann boundary conditions.\nGiven data *u**D*, *f*, *g* as before, the dual weak formulation of seeks (*σ*, *u*) ∈ Σ(*g*) × *L*2(Ω; R2) with\n \n$$\\begin{aligned}\n \\label{eq:lame-weak}\n \\begin{split}\n \\int\\_\\Omega \\sigma:{\\mathbb{C}}^{-1}\\tau\\;dx + \\int\\_\\Omega u\\cdot{\\operatorname{div}}\\tau\\;dx\n &= \\int\\_{\\Gamma\\_D} u\\_D\\cdot(\\tau\\nu)\\,ds\\quad\\textrm{for all }\\tau\\in\\Sigma\\_0,\\\\\n \\int\\_\\Omega v\\cdot{\\operatorname{div}}\\sigma\\;dx &= -\\int\\_\\Omega f\\cdot v\\,dx\\quad\\textrm{for all } v\\in L^2(\\Omega;\\mathbb{R}^2).\n \\end{split}\\end{aligned}$$ \nIt is well known that the two formulations are equivalent and well posed in the sense that they allow for unique solutions in the above spaces and are actually slightly more regular according to the reduced elliptic regularity theory. The reader is refereed to textbooks on finite element methods for proofs and further descriptions.\nThroughout this paper, the model problem considers truly mixed boundary conditions with the hypothesis that both Γ*D* and Γ*N* have positive length. The remaining cases of a pure Neumann problem or a pure Dirichlet problem require standard modification and are immediately adopted. The presentation focusses on the case of isotropic linear elasticity with constant Lamé parameters *λ* and *μ* for brevity and many results carry over to more general situations (cf. Remark [rematkcc1] and [remarkonjumptermvanishes] for instance).\nMixed finite element discretization\n-----------------------------------\nLet T denote a shape-regular triangulation of Ω into triangles (in the sense of Ciarlet ) with set of nodes N, set of interior edges E(Ω), set of Dirichlet edges E(Γ*D*) and set of Neumann edges E(Γ*N*). The triangulation is compatible with the boundary pieces Γ*D* and Γ*N* in that the boundary condition changes only at some vertex N and Γ*D* (resp. $\\overline{\\Gamma\\_N}$) is partitioned in E(Γ*D*) (resp. E(Γ*N*)).\nThe piecewise polynomials (piecewise with respect to the triangulation T) of total degree at most *k* ∈ N0 are denoted as *P**k*(T), their vector- or matrix-valued versions as *P**k*(T; R2) or *P**k*(T; R2 × 2) etc. The subordinated Arnold-Winther finite element space *A**W**k*(T) of index *k* ∈ N reads \n$$AW\\_k({\\mathcal{T}}):=\\big\\{\\tau\\in P\\_{k+2}({\\mathcal{T}};\\mathbb S)\\cap H({\\operatorname{div}},\\Omega;\\mathbb{S})\\,\n \\big|\\,{\\operatorname{div}}\\tau\\in P\\_k({\\mathcal{T}};\\mathbb{R}^2)\\big\\}.$$ \n The Neumann boundary conditions are essential conditions and are traditionally implemented by some approximation *g**h* to *g* in the normal trace space \n$$G({\\mathcal{T}}):=\\{ (\\tau\\_h \\nu) |\\_{\\Gamma\\_N} \\in L^2(\\Gamma\\_N;{\\mathbb{R}}^2) \\,\\big|\\,\n \\tau\\_h\\in AW\\_k({\\mathcal{T}}) \\}$$ \n (recall that *ν* is the exterior unit normal along the boundary). Given any *g**h* ∈ *G*(T), the discrete stress approximations are sought in the non-void affine subspace \nΣ(*g**h*, T) : = Σ(*g*) ∩ *A**W**k*(T) \n of *A**W**k*(T) with test functions in the linear subspace Σ(0, T) : = Σ0 ∩ *A**W**k*(T). Then there exists a unique discrete solution *σ**h* ∈ Σ(*g**h*, T) and *u**h* ∈ *V**h* : = *P**k*(T; R2) to\n \n$$\\begin{aligned}\n\\label{eq:discpp}\n\\begin{split}\n\\int\\_\\Omega\\sigma\\_h :{\\mathbb{C}}^{-1}\\tau\\_h\\,dx+\\int\\_\\Omega u\\_h\\cdot{\\operatorname{div}}\\tau\\_h\\,dx\n& = \\int\\_{\\Gamma\\_D} u\\_D \\cdot(\\tau\\_h\\nu)\\, ds \n\\quad \\textrm{for all } \\tau\\_h\\in\\Sigma(0,{\\mathcal{T}}),\\\\\n\\int\\_\\Omega v\\_{h}\\cdot {\\operatorname{div}}\\sigma\\_h\\, dx & = \n\\int\\_\\Omega f\\cdot v\\_{h}\\, dx\\quad\\textrm{ for all } v\\_{h}\\in V\\_h.\n\\end{split}\\end{aligned}$$ \nThe explicit design of a Fortin projection leads in to quasi-optimal a priori error estimates for an exact solution $(\\sigma, u)\\in(\\Sigma(g)\\cap H^{k+2}(\\Omega;\\mathbb S))\n\\times H^{k+2}(\\Omega)$ to and the approximate solution (*σ**h*, *u**h*) to, namely (with the maximal mesh-size *h*)\n \n$$\\begin{aligned}\n \\|\\sigma-\\sigma\\_h\\|\\_{L^2(\\Omega)}&\\lesssim h^{m}\\|\\sigma\\|\\_{H^m(\\Omega)}\\quad\\mbox{for } 1\\leq m \\leq k+2,\\\\\n \\| u- u\\_h\\|\\_{L^2(\\Omega)}&\\lesssim h^{m}\\|u\\|\\_{H^{m+1}(\\Omega)}\\quad\\mbox{for } 1\\leq m \\leq k+1. \\end{aligned}$$ \nAnother stable pair of different and mesh-depending norms in implies the *L*2 best approximation of the stress error *σ* − *σ**h* up to a generic multiplicative constant and data oscillations on *f* under some extra condition (N) on the Neumann data approximation *g**h* implied by the first and zero moment orthogonality assumption *g* − *g**h* ⊥ *P*1(E(Γ*N*); R2) ( ⊥ indicates orthogonality in *L*2(Γ*N*)) met in all the numerical examples of this paper.\nFor simple benchmark examples with piecewise polynomial data *f* and *g*, there is even a superconvergence phenomenon visible in numerical examples. The arguments of this paper allow a proof of fourth-order convergence of the *L*2 stress error ∥*σ* − *σ**h*∥ = *o*(*h*4) in the lowest-order Arnold-Winther method with *k* = 1 for a smooth stress *σ* ∈ *H*4(Ω; S) with *f* = *f**h* ∈ *P*1(T; R2) and *g* = *g**h* ∈ *G*(T). (In fact, once the data are not piecewise affine, the arising oscillation terms are only of at most third order and the aforementioned convergence estimates are sharp.)\nThis is stated as Theorem [supperconvergence] in the appendix, because the a priori error analysis lies outside of the main focus of this work. It is surprising though that adaptive mesh-refining suggested with this paper recovers this higher convergence rate even for the inconsistent Neumann data in the Cook membrane benchmark example below.\nExplicit residual-based a posteriori error estimator\n----------------------------------------------------\nThe novel explicit residual-based error estimator for the discrete solution (*σ**h*, *u**h*) to depends only on the Green strain approximation C− 1*σ**h* and its piecewise derivatives and jumps across edges.\nGiven any edge *E* of length *h**E*, let *ν**E* denote the unit normal vector (chosen with a fixed orientation such that it points outside along the boundary ∂Ω of Ω) and let *τ**E* denote its tangential unit vector; by convention *τ**E* = (0, − 1; 1, 0)*ν**E* with the indicated asymmetric 2 × 2 matrix. The tangential derivative *τ**E* ⋅ ∇ • along an edge (or boundary) is identified with the one-dimensional derivative ∂ • /∂*s* with respect to the arc-length parameter *s*. The jump [*v*]*E* of any piecewise continuous scalar, vector, or matrix *v* across an interior edge *E* = ∂*T*+ ∩ ∂*T*− shared by the two triangles *T*+ and *T*− such that *ν**E* points outside *T*+ along *E* ⊂ ∂*T*+ reads \n[*v*]*E* : = (*v*∣*T*+)∣*E* − (*v*∣*T*−)∣*E*. \n The rotation acts on a vector field Φ (and row-wise on matrices) via rotΦ : = ∂1Φ2 − ∂2Φ1 and rot*N**C* denotes its piecewise application.\nUnder the present notation and the throughout abbreviation ${\\bm{\\varepsilon}\\_h}:={\\mathbb{C}}^{-1}\\sigma\\_h$, the explicit residual-based a posteriori error estimator reads\n \n$$\\begin{aligned}\n\\label{eq:estimator}\\nonumber\n&\\hspace{-20mm} \n\\displaystyle \\eta^2({\\mathcal{T}},\\sigma\\_h):=\\sum\\_{T\\in{\\mathcal{T}}}h\\_T^4 \\|{\\operatorname{rot}}{\\operatorname{rot}}{\\bm{\\varepsilon}\\_h}\\|\\_{L^2(T)}^2 +{\\operatorname{osc}}^2(f,{\\mathcal{T}})\n+ {\\operatorname{osc}}^2(g-g\\_h,{\\mathcal{E}}(\\Gamma\\_N)) \\\\ &\n\\hspace{-5mm}+\\hspace{-2mm}\\displaystyle\n\\sum\\_{E\\in{\\mathcal{E}}(\\Omega)}\\hspace{-2mm} \\left(\\hspace{-1mm} h\\_E \\|\\tau\\_E \\cdot [{\\bm{\\varepsilon}\\_h}]\\_E \\tau\\_E \\|\\_{L^2(E)}^2\n+ h\\_E^3 \\| \\tau\\_E\\cdot ( [{\\operatorname{rot}}\\_{NC}{\\bm{\\varepsilon}\\_h}]\\_E - \\frac{ \\partial [{\\bm{\\varepsilon}\\_h}]\\_E \\nu\\_E }{\\partial s })\\|\\_{L^2(E)}^2\n\\hspace{-1mm}\\right)\n\\\\ &\\hspace{-1mm} +\\hspace{-3mm}\\displaystyle \\sum\\_{E\\in{\\mathcal{E}}(\\Gamma\\_D)} \n\\hspace{-2mm} \\left(\\hspace{-1mm}h\\_E\\|\\tau\\_E \\cdot ({\\bm{\\varepsilon}\\_h}\\tau\\_E - \\frac{ \\partial u\\_D}{\\partial s}) \\|\\_{L^2(E)}^2 + \nh\\_E^3 \\|\\tau\\_E\\cdot {\\operatorname{rot}}{\\bm{\\varepsilon}\\_h}- \\nu\\_E\\cdot (\\frac{ \\partial {\\bm{\\varepsilon}\\_h}\\tau\\_E }{\\partial s }\n +\\frac{ \\partial^2 u\\_D}{ \\partial s^2}) \\|\\_{L^2(E)}^2\\hspace{-1mm}\\right) \\nonumber\n \\end{aligned}$$ \nfor the oscillations osc(*f*, T) of the volume force and the oscillations osc(*g* − *g**h*, E(Γ*N*)) of the traction boundary condition, defined below.\n[reliability][thmREL] There exists a mesh-size and *λ* independent constant *C*rel (which may depend on *μ* and on the shape-regularity of the triangulation T through a global lower bound of the minimal angle therein) such that the exact (resp. discrete) stress *σ* from (resp. *σ**h* from with *g* − *g**h* ⊥ *P*0(E(Γ*N*); R2) and the error estimator satisfy\n \n$$\\begin{aligned}\n \\| \\sigma - \\sigma\\_h \\|\\_{L^2(\\Omega)} \\leq C\\_{\\textrm{rel}} \\eta({\\mathcal{T}},\\sigma\\_h).\\end{aligned}$$ \nThe a posteriori error estimator *η*(T, *σ**h*) already involves two data oscillation terms osc(*f*, T) and osc(*g* − *g**h*, E(Γ*N*)) defined as the square roots of the respective terms in\n \n$$\\begin{aligned}\n&{\\operatorname{osc}}^2(f,{\\mathcal{T}}):= \\sum\\_{T\\in{\\mathcal{T}}} h\\_T^2\\|f-f\\_h\\|\\_{L^2(T)}^2 \\text{ for the }L^2\\text{ projection }f\\_h \\text{ of } f\n\\text{ onto } P\\_k({\\mathcal{T}};{\\mathbb{R}}^2); \\\\\n&{\\operatorname{osc}}^2(g-g\\_h,{\\mathcal{E}}(\\Gamma\\_N)):= \\sum\\_{E\\in{\\mathcal{E}}(\\Gamma\\_N)} h\\_E \\|g-g\\_h\\|\\_{L^2(E)}^2.\\end{aligned}$$ \nFor any edge *E* and a degree *m* ≥ *k* + 2, let Π*m*, *E* : *L*2(*E*) → *P**m*(*E*) denote the *L*2 projection onto polynomials of degree at most *m*. For any *E* ∈ E(Γ*D*) define the two Dirichlet data oscillation terms\n \n$$\\begin{aligned}\n\\label{eqdefoscIondirichlet}\n{\\operatorname{osc}}^2\\_I(u\\_D,E)&:=& h\\_E\\| (1-\\Pi\\_{m,E}) \\partial (u\\_D\\cdot\\tau\\_E)/\\partial s \\|\\_{L^2(E)}^2, \\\\\n\\label{eqdefoscIIondirichlet}\n{\\operatorname{osc}}^2\\_{II}(u\\_D,E)&:= &\nh\\_E^3\\| (1-\\Pi\\_{m,E}) \\partial^2 ( u\\_D\\cdot\\nu\\_E )/ \\partial s^2\\|\\_{L^2(E)}^2.\\end{aligned}$$ \nTheir sum defines the overall Dirichlet data approximation osc(*u**D*, E(Γ*D*)) as the square root of \nosc2(*u**D*, E(Γ*D*)) : = ∑*E* ∈ E(Γ*D*)(osc*I*2(*u**D*, *E*) + osc*I**I*2(*u**D*, *E*)). \nThe analysis of Section [sec:efficiency] is local and states for each of the five local residuals an upper bound related to the error in a neighborhood. The global efficiency is displayed as follows.\n[efficiency][thmEFF] There exists a mesh-size and *λ*, *μ* independent constant *C*eff (which may depend on the shape-regularity of the triangulation T through a global lower bound of the minimal angle therein) such that the exact (resp. discrete) stress *σ* from (resp. *σ**h* from with *g* − *g**h* ⊥ *P*0(E(Γ*N*); R2) and the error estimator satisfy\n \n$$\\begin{aligned}\nC\\_{\\textrm{eff}}^{-1} \\eta({\\mathcal{T}},\\sigma\\_h) \\leq \\| \\sigma - \\sigma\\_h \\|\\_{L^2(\\Omega)}\n+ {\\operatorname{osc}}(f,{\\mathcal{T}})+{\\operatorname{osc}}(g-g\\_h,{\\mathcal{E}}(\\Gamma\\_N))+{\\operatorname{osc}}(u\\_D,{\\mathcal{E}}(\\Gamma\\_D)).\\end{aligned}$$ \nOutline of the paper\n--------------------\nThe remaining parts of this paper provide a mathematical proof of Theorem [thmREL] and Theorem [thmEFF] and numerical evidence in computational experiments on the novel a posteriori error estimation and its robustness as well as on associated mesh-refining algorithms.\nThe proof of the reliability of Theorem [thmREL] in Section [sec:reliability] adopts arguments of and carries out two integration by parts on each triangle plus one-dimensional integration by parts along all edges. The resulting terms are in fact locally efficient in Section [sec:efficiency] with little generalizations of the bubble-function methodology due to Verfürth. The five lemmas of Section [sec:efficiency] give slightly sharper results and in total imply Theorem [thmEFF].\nThe point in Theorem [thmREL] and [thmEFF] is that the universal constants *C*rel and *C*eff may depend on the Lamé parameter *μ* but are independent of the critical Lamé parameter *λ* as supported by the benchmark examples of the concluding Section [sec:examples]. Adaptive mesh-refining proves to be highly effective with the novel a posteriori error estimator even for incompatible Neumann data. Four benchmark examples with the Poisson ratio *ν* = 0.3 or 0.4999 provide numerical evidence of the robustness of the reliabile and efficient a posteriori error estimation and for the fourth-order convergence of Theorem [supperconvergence].\nComments on general notation\n----------------------------\nStandard notation on Lebesgue and Sobolev spaces and norms is adopted throughout this paper and, for brevity, ∥ ⋅ ∥ : = ∥ ⋅ ∥*L*2(Ω) denotes the *L*2 norm. The piecewise action of a differential operator is denoted with a subindex *N**C*, e.g., ∇*N**C* denotes the piecewise gradient (∇*N**C* • )∣*T* : = ∇( • ∣*T*) for all *T* ∈ T. Sobolev functions are usually defined on open sets and the notation *W**m*, *p*(*T*) (resp. *W**m*, *p*(T)) substitutes *W**m*, *p*(int(*T*)) for a (compact) triangle *T* and its interior int(*T*) (resp. *W**m*, *p*(int(T))) and their vector and matrix versions.\nFor a differentiable function *ϕ*, Curl*ϕ* : = ( − ∂2*ϕ*, ∂1*ϕ*) is the rotated gradient; for a two-dimensional vector field Φ, CurlΦ is the 2 × 2 matrix-valued rotated gradient \nCurlΦ : = ( − ∂2Φ1, ∂1Φ1; − ∂2Φ2, ∂1Φ2) = *D*Φ(0, 1; − 1, 0). \n (The signs are not uniquely determined in the literature and some care is required.)\nThe colon denotes the scalar product *A* : *B* : = ∑*α*, *β* = 1, 2*A**α*, *β**B**α*, *β* of 2 × 2 matrices *A*, *B*. The inequality $A\\lesssim B$ between two terms *A* and *B* abbreviates *A* ≤ *C* *B* with some multiplicative generic constant *C*, which is independent of the mesh-size and independent of the one Lamé parameter *λ* ≥ 0 but may depend on the other *μ* > 0 and may depend on the shape-regularity of the underlying triangulation T.\nProof of reliability\n====================\nThis section is devoted to the proof of Theorem [thmREL] based on a Helmholtz decomposition on with two parts as decomposed in Theorem [thm:Helmholtz] below. The critical part is the *L*2 product of C− 1(*σ* − *σ**h*) times the Curl of an unknown function Curl*β*. The observation from is that one can find an Argyris finite element approximation *β**h* to *β* ∈ *H*2(Ω) such that the continuous function *ϕ* : = *β* − *β**h* ∈ *H*2(Ω) vanishes at all vertices N of the triangulation. Two integration by parts on each triangle plus one-dimensional integration by parts along the edges E of the triangulation eventually lead to a key identity.\n[representation formula][lemma:representation] Any function ${\\bm{\\varepsilon}\\_h}\\in H^2({\\mathcal{T}};\\mathbb{S})$ (i.e. ${\\bm{\\varepsilon}\\_h}$ is piecewise in *H*2 with values in S) and any *ϕ* ∈ *H*2(Ω) with *ϕ*(*z*) = 0 at all vertices *z* ∈ N in the regular triangulation T satisfy\n \n$$\\begin{aligned}\n& \\hspace{-20mm} ({\\bm{\\varepsilon}\\_h}, {\\operatorname{Curl}}^2 \\phi)\\_{L^2(\\Omega)} =({\\operatorname{rot}}\\_{NC}{\\operatorname{rot}}\\_{NC}{\\bm{\\varepsilon}\\_h},\\phi)\\_{L^2(\\Omega)}\n \\\\ & \\displaystyle +\\sum\\_{E\\in{\\mathcal{E}}(\\Omega)} \\left( (\\tau\\_E\\cdot[{\\bm{\\varepsilon}\\_h}]\\_E\\tau\\_E,\\partial\\_{\\nu\\_E}\\phi)\\_{L^2(E)}\n - ([{\\operatorname{rot}}\\_{NC}{\\bm{\\varepsilon}\\_h}]\\_E - \\frac{ \\partial [{\\bm{\\varepsilon}\\_h}]\\_E\\nu\\_E}{\\partial s},\\phi \\, \\tau\\_E )\\_{L^2(E)} \\right) \n \\\\& \\displaystyle +\\sum\\_{E\\in{\\mathcal{E}}(\\partial\\Omega)} \\left( (\\tau\\_E\\cdot{\\bm{\\varepsilon}\\_h}\\tau\\_E,\\partial\\_{\\nu\\_E}\\phi )\\_{L^2(E)}\n - ({\\operatorname{rot}}{\\bm{\\varepsilon}\\_h}-\\frac{ \\partial {\\bm{\\varepsilon}\\_h}\\nu\\_E}{\\partial s}, \\phi \\, \\tau\\_E)\\_{L^2(E)}\\right).\\end{aligned}$$ \nThe subsequent integration by parts formula is utilized frequently throughout this paper for *ϕ* ∈ *H*1(Ω; R2) and Ψ ∈ *H*1(Ω; R2 × 2) \n∫ΩΨ : Curl*ϕ* *d**x* + ∫Ω*ϕ* ⋅ rotΨ *d**x* = ∫∂Ω*ϕ* ⋅ Ψ*τ**E* *d**s*. \n Any differentiable (scalar) function *φ*, satisfies the elementary relations \n*τ**E* ⋅ Curl*φ* = ∂*φ*/∂*ν**E* and *ν**E* ⋅ Curl*φ* = − ∂*φ*/∂*s* = − ∂*φ*/∂*τ**E* on *E* ∈ E. \nIntegrate by parts twice on each triangle and rearrange the remaining boundary terms to deduce (with the abbreviation rot*N**C*rot*N**C* ≡ rot*N**C*2)\n \n$$\\begin{aligned}\n&&\\hspace{-10mm} ({\\bm{\\varepsilon}\\_h},{\\operatorname{Curl}}^2\\phi)\\_{L^2(\\Omega)} = ({\\operatorname{rot}}\\_{NC}^2{\\bm{\\varepsilon}\\_h},\\phi)\\_{L^2(\\Omega)} \\\\\n&&+\\sum\\_{E\\in{\\mathcal{E}}(\\Omega)} \\left( ([{\\bm{\\varepsilon}\\_h}]\\_E\\tau\\_E,{\\operatorname{Curl}}\\phi)\\_{L^2(E)}\n-([{\\operatorname{rot}}\\_{NC}{\\bm{\\varepsilon}\\_h}]\\_E\\cdot\\tau\\_E,\\phi )\\_{L^2(E)} \\right) \\\\ \n&&+\\sum\\_{E\\in{\\mathcal{E}}(\\partial \\Omega)} \\left( ({\\bm{\\varepsilon}\\_h}\\tau\\_E,{\\operatorname{Curl}}\\phi)\\_{L^2(E)}\n-({\\operatorname{rot}}{\\bm{\\varepsilon}\\_h}\\cdot\\tau\\_E,\\phi )\\_{L^2(E)} \\right).\\end{aligned}$$ \nThe term $ ([{\\bm{\\varepsilon}\\_h}]\\_E\\tau\\_E,{\\operatorname{Curl}}\\phi)\\_{L^2(E)} $ in the above sum is split into orthogonal components \nCurl*ϕ* = (*τ**E* ⋅ Curl*ϕ*)*τ**E* + (*ν**E* ⋅ Curl*ϕ*)*ν**E* = (*τ**E* ⋅ Curl*ϕ*)*τ**E* − (∂*ϕ*/∂*s*)*ν**E* on *E* ∈ E. \n Since *ϕ* vanishes at the vertices, an integration by parts along each interior edge *E* for the last term shows $ ([{\\bm{\\varepsilon}\\_h}]\\_E\\tau\\_E, (\\partial \\phi/\\partial s )\\nu\\_E)\\_{L^2(E)} = - (\\partial [{\\bm{\\varepsilon}\\_h}]\\_E\\tau\\_E/\\partial s, \\phi\\nu\\_E)\\_{L^2(E)}$. This proves \n$$([{\\bm{\\varepsilon}\\_h}]\\_E\\tau\\_E,{\\operatorname{Curl}}\\phi)\\_{L^2(E)} = (\\tau\\_E\\cdot [{\\bm{\\varepsilon}\\_h}]\\_E\\tau\\_E,\\partial\\_{\\nu\\_E}\\phi)\\_{L^2(E)} +\n (\\frac{\\partial \\nu\\_E\\cdot[{\\bm{\\varepsilon}\\_h}]\\_E\\tau\\_E}{\\partial s},\\phi)\\_{L^2(E)}.$$ \n The same formula holds for any boundary edge *E* when $[{\\bm{\\varepsilon}\\_h}]\\_E$ is replaced by ${\\bm{\\varepsilon}\\_h}$. The combination of the latter identities with the first displayed formula of this proof verifies the asserted representation formula.\nThe contribution of *ε*(*u*) = C− 1*σ* times the Curl2*ϕ* ∈ *L*2(Ω; S) exclusively leads to boundary terms. Throughout this paper, suppose that the Dirichlet data *u**D* satisfies *u**D* ∈ *C*(Γ*D*) ∩ *H*2(E(Γ*D*) in the sense that *u**D* is globally continuous with *u**D*∣*E* ∈ *H*2(*E*; R2) for all *E* ∈ E(Γ*D*).\n[boundary terms][lemma:bd] Any Sobolev function *v* ∈ *H*1(Ω; R2) with boundary values *u**D* ∈ *C*(Γ*D*) ∩ *H*2(E(Γ*D*) on Γ*D* and any *ϕ* ∈ *H*2(Ω) with *ϕ* = ∂*ϕ*/∂*ν* = 0 along Γ*N* with *ϕ*(*z*) = 0 for any vertex *z* of Γ*D* in its relative interior satisfy \n$$(\\varepsilon (v), {\\operatorname{Curl}}^2 \\phi)\\_{L^2(\\Omega)}=\\sum\\_{E\\in{\\mathcal{E}}(\\Gamma\\_D)} \n \\left( (\\frac{ \\partial u\\_D}{\\partial s }, \\frac{ \\partial\\phi}{\\partial \\nu\\_E}\\, \\tau\\_E )\\_{L^2(E)}\n + (\\frac{\\partial^2 u\\_D}{\\partial s^2 },\\phi\\, \\nu\\_E )\\_{L^2(E)}\\right).$$ \nA density argument shows that it suffices to prove this identity for smooth functions *v* and *ϕ*, when integration by parts arguments show that the left-hand side is equal to \n$$\\int\\_{\\partial\\Omega} {\\operatorname{Curl}}\\phi \\cdot \\frac{\\partial v }{\\partial s}ds=\n\\sum\\_{E\\in {\\mathcal{E}}(\\partial\\Omega)}\n\\int\\_{E} \\left( \\frac{\\partial\\phi}{\\partial \\nu\\_E} \\frac{\\partial (v\\cdot\\tau\\_E)}{\\partial s}\n+ \\phi \\frac{\\partial^2 (v\\cdot \\nu\\_E) }{\\partial s^2}\\right)ds.$$ \n The equality follows from an orthogonal split Curl*ϕ* = (*τ* ⋅ Curl*ϕ*)*τ* + (*ν* ⋅ Curl*ϕ*)*ν* into the normal and tangential directions of *ν* and *τ* along the boundary ∂Ω followed by an integration by parts along ∂Ω with *ϕ*(*z*) = 0 for vertices *z* in Γ*D* with a jump of the normal unit vector. The substitution of the boundary conditions concludes the proof.\nThe consequence of the previous two lemmas is a representation formula for the error times a typical function Curl2*ϕ*. To understand why the contributions on the Neumann boundary of *ϕ* and ∇*ϕ* disappear along Γ*N*, some details on the Helmholtz decomposition are recalled from the literature. For this, let Γ0, …, Γ*J* denote the compact connectivity components of $\\overline{\\Gamma\\_N}$.\n[Helmholtz decomposition ] [thm:Helmholtz] Given any *σ* − *σ**h* ∈ *L*2(Ω; S), there exists *α* ∈ *V*, constant vectors *c*0, …, *c**J* ∈ R2 with *c*0 = 0 and *β* ∈ *H*2(Ω) with ∫Ω*β* *d**x* = 0 and Curl*β* = *c**j* on Γ*j* ⊆ Γ*N* for all *j* = 0, …, *J* such that\n \n$$\\begin{aligned}\n\\label{e:helmholtz}\n \\sigma-\\sigma\\_h = {\\mathbb{C}}\\varepsilon(\\alpha) + {\\operatorname{Curl}}{\\operatorname{Curl}}\\beta. \\qed\n \\end{aligned}$$ \nThe second ingredient is an approximation *β**h* of *β* from the Helmholtz decomposition in Theorem [thm:Helmholtz] based on the Argyris finite element functions *A*(T) ⊂ *C*1(Ω) ∩ *P*5(T). The local mesh-size *h*T ∈ *P*0(T) in the triangulation T is defined as its diameter *h*T∣*T* : = *h**T* on each triangle *T* ∈ T.\n[quasiinterpolation][lemma:quasi] Given any *β* as in Theorem [thm:Helmholtz] there exists some *β**h* ∈ *A*(T) such that *ϕ* : = *β* − *β**h* ∈ *H*2(Ω) vanishes at any vertex *z* ∈ N of the triangulation, *ϕ* and its gradient ∇*ϕ* vanish on Γ*N*, and the local approximation and stability property holds in the sense that \n$$\\| h\\_{\\mathcal{T}}^{-2} \\phi \\| + \\| h\\_{\\mathcal{T}}^{-1}{\\operatorname{Curl}}\\phi \\| +\\| {\\operatorname{Curl}}^2\\phi \\| \\lesssim \\| \\beta \\|\\_{H^2(\\Omega)}.$$ \nThis has been (partly) utilized in and also follows from.\nThe combination of all aforementioned arguments leads to the following estimate as an answer to the question of Subsection [subsec:1.1] in terms of directional derivatives of ${\\bm{\\varepsilon}\\_h}:= {\\mathbb{C}}^{-1}\\sigma\\_h$. Recall the definition of *η*(T, *σ**h*) from.\n[key result][thm:keyresult] Let *σ* ∈ *H*(div, Ω; S) solve and let *σ**h* ∈ *A**W**k*(T) solve. Given *β* from Theorem [thm:Helmholtz] and its quasiinterpolation *β**h* from Lemma [lemma:quasi], the difference *ϕ* : = *β* − *β**h* satisfies \n$$({\\mathbb{C}}^{-1}(\\sigma-\\sigma\\_h), {\\operatorname{Curl}}^2\\phi)\\_{L^2(\\Omega)} \\lesssim | \\beta |\\_{H^2(\\Omega)} \\eta({\\mathcal{T}},\\sigma\\_h).$$ \nLemma [lemma:representation] and Lemma [lemma:bd] lead to a formula for $({\\bm{\\varepsilon}\\_h}, {\\operatorname{Curl}}^2\\phi)\\_{L^2(\\Omega)}$, ${\\bm{\\varepsilon}\\_h}:= {\\mathbb{C}}^{-1}\\sigma\\_h$, in which all the contributions for *E* ∈ E(Γ*N*) with *ϕ* and ∇*ϕ* vanish along Γ*N*. The remaining formula reads\n \n$$\\begin{aligned}\n &({\\mathbb{C}}^{-1}(\\sigma-\\sigma\\_h),{\\operatorname{Curl}}^2\\phi)\\_{L^2(\\Omega)}\n = - ({\\operatorname{rot}}\\_{NC}^2{\\bm{\\varepsilon}\\_h},\\phi)\\_{L^2(\\Omega)}\n \\\\ & \\displaystyle -\\sum\\_{E\\in{\\mathcal{E}}(\\Omega)} \\left( \n (\\tau\\_E\\cdot[{\\bm{\\varepsilon}\\_h}]\\_E\\tau\\_E,\\frac{\\partial \\phi }{\\partial \\nu\\_E})\\_{L^2(E)}\n - ([{\\operatorname{rot}}\\_{NC}{\\bm{\\varepsilon}\\_h}]\\_E - \\frac{ \\partial [{\\bm{\\varepsilon}\\_h}]\\_E\\nu\\_E}{\\partial s},\\phi \\, \\tau\\_E )\\_{L^2(E)} \\right) \n \\\\& +\\sum\\_{E\\in{\\mathcal{E}}(\\Gamma\\_D)} \\left( \n ( \\frac{\\partial u\\_D}{\\partial s} - {\\bm{\\varepsilon}\\_h}\\tau\\_E, \\tau\\_E\\, \\frac{\\partial\\phi}{\\nu\\_E} )\\_{L^2(E)} \\right.\n \\\\&\\qquad \\quad\\left. + ( \\tau\\_E\\cdot ( {\\operatorname{rot}}\\_{NC}{\\bm{\\varepsilon}\\_h}-\\frac{\\partial( {\\bm{\\varepsilon}\\_h}\\nu\\_E)}{\\partial s} )\n +\\frac {\\partial^2 u\\_D\\cdot\\nu\\_E}{\\partial s^2 },\\phi )\\_{L^2(E)}\\right).\n \\end{aligned}$$ \nThe proof concludes with Cauchy-Schwarz inequalities, trace inequalities, and the approximation estimates of Lemma [lemma:quasi]. The remaining details are nowadays standard arguments in the a posteriori error analysis of nonconforming and mixed finite element methods and hence are omitted.\nBefore the proof of Theorem [thmREL] concludes this section, three remarks and one lemma are in order.\n[nonconstant coefficients][rematkcc1] The main parts of the reliability analysis of this section hold for rather general material tensors C as long as ${\\bm{\\varepsilon}\\_h}:= {\\mathbb{C}}^{-1}\\sigma\\_h$ allows for the existence of the traces and the derivatives in the error estimator in the respective *L*2 spaces. For instance, if *λ* and *μ* are piecewise smooth with respect to the underlying triangulation T.\n[constant coefficients][remarkonjumptermvanishes] The overall assumption of constant Lamé parameters *λ* and *μ* allows a simplification in the error estimator. It suffices to have *μ* globally continuous and *μ* and *λ* piecewise smooth to guarantee \n$$\\frac{ \\partial [{\\bm{\\varepsilon}\\_h}]\\_E\\nu\\_E}{\\partial s} \\cdot \\tau\\_E =0\\quad\\text{along }E\\in{\\mathcal{E}}(\\Omega).$$ \n (The proof utilizes the structure of C and C− 1 with ${\\mathbb{C}}^{-1} E = \\frac{1}{2\\mu} (E - \\frac{\\lambda}{2(\\lambda+\\mu)} {\\operatorname{tr}}(E) 1\\_{2\\times 2})$ for any *E* ∈ S as a linear combination of the identity and some scalar multiple of the 2 × 2 unit matrix. The terms with the identity lead to 1/(2*μ*) times the jump [*σ**h*]*E**ν**E* = 0 of the *H*(div) conforming stress approximations. The jump terms with the unit matrix (even with jumps of *λ*) are multiplied with the orthogonal unit vectors *ν**E* and *τ**E* and so vanish as well.)\n[Related work] Although the work concerns a different problem (bending of a plate of fourth order) with a different discretisation (even nonconforming in *H*(div)), some technical parts of that paper are related to those of this by a rotation of the underlying coordinate system and the substitution of divdiv by rotrot etc. Another Helmholtz decomposition also allows for a discrete version and thereby enables a proof of optimal convergence of an adaptive algorithm with arguments from.\nA technical detail related to the robustness in *λ* → ∞ is a well known lemma that controls the trace of a matrix *E* ∈ R2 × 2 by its deviatoric part dev*E* : = *E* − tr(*E*)/2 12 × 2 and its divergence measured in the dual *V*\\* ⊂ *H*− 1(Ω; R2) of *V*, namely \n$$\\| {\\operatorname{div}}\\tau \\|\\_{-1} := \\sup\\_{\\substack{v\\in V \\\\ | v |\\_{H^1(\\Omega)} =1}}{\\int\\_\\Omega \\tau: Dv\\, dx}\n\\quad\\text{for all } \n\\tau\\in L^2(\\Omega;{\\mathbb{R}}^{2\\times 2}).$$ \n[tr-dev-div][l:trdevdivCCDolzmann] Let Σ0 be a closed subspace of $H({\\operatorname{div}},\\Omega;\\mathbb R^{2\\times 2})$, which does not contain the constant tensor 12 × 2. Then any *τ* ∈ Σ0 satisfies\n \n$$\\begin{aligned}\n\\|\\operatorname{tr}(\\tau)\\|\\_{L^2(\\Omega)}\n \\lesssim \\|\\operatorname{dev}\\tau\\|\\_{L^2(\\Omega)} + \\|{\\operatorname{div}}\\tau\\|\\_{-1}.\\end{aligned}$$ \nThere are several variants of the tr-dev-div lemma known in the literature [BF91, Proposition 7 in Section IV.3]. The version in explicitly displays a version with ∥div*τ*∥ replacing ∥div*τ*∥− 1. Since its proof is immediately adopted to prove the asserted version, further details are omitted.\nThe remaining part of this section outlines why Theorem [thmREL] follows from Theorem [thm:keyresult] with the arguments from. The energy norms of *v* ∈ *V* and *τ* ∈ *H*(div, Ω; S) read \n| | |*v*| | |2 : = ∫Ω*ɛ*(*v*) : C*ɛ*(*v*) *d**x* and ∥*τ*∥C− 12 : = ∫Ω*τ* : C− 1*τ* *d**x*. \n The remaining residual\n \n$$\\begin{aligned}\n \\operatorname{Res}(v) := \\int\\_\\Omega f\\cdot v\\, dx + \\int\\_{\\Gamma\\_N} g\\cdot v\\, ds \n - \\int\\_\\Omega \\sigma\\_h:\\varepsilon(v)\\, dx\\quad\\text{for all }v\\in V \\end{aligned}$$ \nwith its dual norm\n \n$$\\begin{aligned}\n{\\lvert\\!\\lvert\\!\\lvert\\operatorname{Res}\\rvert\\!\\rvert\\!\\rvert}\\_\\* := \\sup\\_{\\substack{v\\in V \\\\ {\\lvert\\!\\lvert\\!\\lvertv \\rvert\\!\\rvert\\!\\rvert}=1}}{\\operatorname{Res}(v)}\\end{aligned}$$ \nallow the following error decomposition, with the weighted *L*2 norms ∥ • ∥C− 1 : = ∥C− 1/2 • ∥*L*2(Ω) and ∥ • ∥C : = ∥C1/2 • ∥*L*2(Ω),\n \n$$\\begin{aligned}\n \\label{e:errorsplit}\n\\|\\sigma-\\sigma\\_h\\|\\_{{\\mathbb{C}}^{-1}}^2 \n= {\\lvert\\!\\lvert\\!\\lvert\\operatorname{Res}\\rvert\\!\\rvert\\!\\rvert}\\_{\\*}^2 + \n\\min\\_{w\\in u\\_D+V}\\|{\\mathbb{C}}^{-1}\\sigma\\_h-\\varepsilon(w)\\|\\_{\\mathbb{C}}^2.\\end{aligned}$$ \nIn fact, it is shown in the proof of that *α* ∈ *V* and *β* ∈ *H*2(Ω) from the Helmholtz decomposition of the error *σ* − *σ**h* in Theorem [thm:Helmholtz] are orthogonal with respect to the *L*2 scalar product weighted with C and that the last term in equals ∥Curl2*β*∥C− 12 = (C− 1(*σ* − *σ**h*), Curl2*ϕ*)*L*2(Ω) and hence is controlled in the key estimate of Theorem [thm:keyresult].\nLemma [l:trdevdivCCDolzmann] applies to Σ0 as the subspace of all *τ* ∈ *H*(div, Ω; S) with homogeneous Neumann data *τ**ν* = 0 along Γ*N*. Since *τ* = Curl2*β* is divergence free and since *τ**ν* = − ∂Curl*β*/∂*s* along Γ*N* (owing to the aforementioned elementary relations and the convention that the first Curl acts row-wise on Curl*β*), where Curl*β* in Theorem [thm:Helmholtz] is piecewise constant, it follows that *τ* ∈ Σ0. On the other hand $1\\_{2\\times 2}\\notin \\Sigma\\_0$ because Γ*N* ≠ ∅. Consequently, Lemma [l:trdevdivCCDolzmann] implies $\\| {\\operatorname{Curl}}^2 \\beta \\|\\lesssim \\| \\operatorname{dev} {\\operatorname{Curl}}^2 \\beta \\|$. This and elementary calculations with C− 1 lead to \n$$| \\beta |\\_{H^2(\\Omega)} = \\| {\\operatorname{Curl}}^2 \\beta \\|\\lesssim \\| \\operatorname{dev} {\\operatorname{Curl}}^2 \\beta \\|\n \\lesssim \\| {\\operatorname{Curl}}^2 \\beta\\|\\_{{\\mathbb{C}}^{-1}}.$$ \n The combination with the estimate resulting from Theorem [thm:keyresult] proves \n$$\\min\\_{w\\in u\\_D+V}\\|{\\mathbb{C}}^{-1}\\sigma\\_h-\\varepsilon(w)\\|\\_{\\mathbb{C}}^2 \n= \\| {\\operatorname{Curl}}^2\\beta\\|\\_{{\\mathbb{C}}^{-1}}^2\n\\lesssim \\| {\\operatorname{Curl}}^2\\beta\\|\\_{{\\mathbb{C}}^{-1}}\\, \\eta({\\mathcal{T}},\\sigma\\_h).$$ \n This implies $\\| {\\operatorname{Curl}}^2\\beta\\|\\_{{\\mathbb{C}}^{-1}}\\lesssim \\eta({\\mathcal{T}},\\sigma\\_h) $ and leads in to\n \n$$\\label{eq:ccreliabilirtyyproof1}\n\\|\\sigma-\\sigma\\_h\\|\\_{{\\mathbb{C}}^{-1}} \\lesssim {\\lvert\\!\\lvert\\!\\lvert\\operatorname{Res}\\rvert\\!\\rvert\\!\\rvert}\\_{\\*} + \\eta({\\mathcal{T}},\\sigma\\_h).$$ \nThe remaining term is the estimate of the dual norm | | |Res| | |\\* of the residual which is done, e.g., in (under the assumption *g* − *g**h* ⊥ *P*0(E(Γ*N*))) \n$${\\lvert\\!\\lvert\\!\\lvert\\operatorname{Res}\\rvert\\!\\rvert\\!\\rvert}\\_{\\*} \\lesssim {\\operatorname{osc}}(f,{\\mathcal{T}})+ {\\operatorname{osc}}(g-g\\_h,{\\mathcal{E}}(\\Gamma\\_N))\\le \\eta({\\mathcal{T}},\\sigma\\_h)$$ \n This and imply \n$$\\|\\operatorname{dev}(\\sigma-\\sigma\\_h)\\|\\lesssim\n\\|\\sigma-\\sigma\\_h\\|\\_{{\\mathbb{C}}^{-1}} \\lesssim \\eta({\\mathcal{T}},\\sigma\\_h).$$ \n For any test function *v* ∈ *V* with ∣*v*∣*H*1(Ω) = 1, ∫Ω(*σ* − *σ**h*) : *D**v* *d**x* = Res(*v*) and so \n$$\\|\\operatorname{div}(\\sigma-\\sigma\\_h)\\|\\_{-1} = \n \\sup\\_{\\substack{v\\in V \\\\ | v |\\_{H^1(\\Omega)} =1}}{Res(v)}\n \\le \\sup\\_{\\substack{v\\in V \\\\ \\| \\epsilon(v) \\| =1}}{Res(v)} \\le 2\\mu\\, {\\lvert\\!\\lvert\\!\\lvert\\operatorname{Res}\\rvert\\!\\rvert\\!\\rvert}\\_{\\*}\n \\lesssim \\eta({\\mathcal{T}},\\sigma\\_h).$$ \n (In the second last step one utilizes that 2*μ* *E* : *E* ≤ *E* : C*E* for all *E* ∈ S.) The combination of Lemma [l:trdevdivCCDolzmann] for *τ* = *σ* − *σ**h* with the previous displayed estimates concludes the proof of $\\|\\sigma-\\sigma\\_h\\|\\lesssim \\eta({\\mathcal{T}},\\sigma\\_h)$. There exist several appropriate choices of Σ0 ⊂ *H*(div, Ω; S) in this last step. Recall that Γ*N* is the union of connectivity components and so pick one edge *E*0 in this polygon and consider Σ0 : = {*τ* ∈ *H*(div, Ω; S) : ∫*E*0*τ**ν* *d**s* = 0} with $1\\_{2\\times 2}\\notin \\Sigma\\_0$. This choice of *E*0 and so Σ0 depend only on Γ*N* (independent of T). Since *g* − *g**h* = (*σ* − *σ**h*)*ν* along *E*0 has (piecewise on E(*E*0), whence in total) an integral mean zero, Lemma [l:trdevdivCCDolzmann] indeed applies to *τ* = *σ* − *σ**h* ∈ Σ0.\nLocal efficiency analysis\n=========================\nThe local efficiency follows with the bubble-function technique for *C*1 finite elements. This section focusses on a constant C for linear isotropic elasticity with constant Lamé parameters *λ* and *μ* such that ${\\bm{\\varepsilon}\\_h}:={\\mathbb{C}}^{-1}\\sigma\\_h\\in P\\_{k+2}({\\mathcal{T}})$ for some *σ**h* ∈ *A**W**k*(T) is a polynomial of degree at most *k* + 2. Appart from this, the Lamé parameters do not further arise in this section.\nThe moderate point of departure is the volume term for each triangle *T* ∈ T with barycentric coordinates *λ*1, *λ*2, *λ*3 ∈ *P*1(*T*) and their product, the cubic volume bubble function, *b**T* : = 27 *λ*1*λ*2*λ*3 ∈ *W*01, ∞(*T*) plus its square *b**T*2 ∈ *W*02, ∞(*T*) with 0 ≤ *b**T*2 ≤ 1, $ \\|b\\_T\\|\\_{L^2(T)} \\lesssim 1$, and $|b\\_T|\\_{H^2(T)} \\lesssim h\\_T^{-2}$ etc.\n[efficiency of volume residual][lemma:eff:1] Any *v* ∈ *H*1(*T*; R2), *T* ∈ T, satisfies\n \n$$\\begin{aligned}\nh\\_T^2 \\| {\\operatorname{rot}}{\\operatorname{rot}}{\\bm{\\varepsilon}\\_h}\\|\\_{L^2(T)} \\lesssim \\| {\\bm{\\varepsilon}\\_h}- \\varepsilon(v) \\|\\_{L^2(T)}.\\end{aligned}$$ \nAn inverse estimate for the polynomial ${\\operatorname{rot}}{\\operatorname{rot}}{\\bm{\\varepsilon}\\_h}\\equiv {\\operatorname{rot}}^2{\\bm{\\varepsilon}\\_h}$ implies the estimate \n$$\\| {\\operatorname{rot}}^2 {\\bm{\\varepsilon}\\_h}\\|\\_{L^2(T)}^2 \\lesssim \\| b\\_T^{1/2} {\\operatorname{rot}}^2 {\\bm{\\varepsilon}\\_h}\\|^2\\_{L^2(T)}\n = ({\\operatorname{rot}}^2{\\bm{\\varepsilon}\\_h}, b\\_T {\\operatorname{rot}}^2 {\\bm{\\varepsilon}\\_h})\\_{L^2(T)}.$$ \n Lemma [lemma:representation] with $\\phi=b\\_T {\\operatorname{rot}}^2 {\\bm{\\varepsilon}\\_h}$ and (*ɛ*(*v*), Curl2*ϕ*)*L*2(*T*) = 0 lead to\n \n$$\\begin{aligned}\n\\begin{split}\n\\| b\\_T^{1/2} {\\operatorname{rot}}^2 {\\bm{\\varepsilon}\\_h}\\|^2\\_{L^2(T)}\n&= ( {\\bm{\\varepsilon}\\_h}- \\varepsilon(v), {\\operatorname{Curl}}^2(b\\_T {\\operatorname{rot}}^2 {\\bm{\\varepsilon}\\_h}) )\\_{L^2(T)}\\\\\n&\\leq \\| {\\bm{\\varepsilon}\\_h}- \\varepsilon(v) \\|\\_{L^2(T)} \\|{\\operatorname{Curl}}^2(b\\_T {\\operatorname{rot}}^2 {\\bm{\\varepsilon}\\_h})\\|\\_{L^2(T)}.\n\\end{split}\\end{aligned}$$ \nThis and the inverse estimate $\\|{\\operatorname{Curl}}^2(b\\_T {\\operatorname{rot}}^2 {\\bm{\\varepsilon}\\_h})\\|\\_{L^2(T)} \\lesssim h\\_T^{-2} \\|b\\_T {\\operatorname{rot}}^2 {\\bm{\\varepsilon}\\_h}\\|\\_{L^2(T)} $ imply \n$$\\| {\\operatorname{rot}}^2 {\\bm{\\varepsilon}\\_h}\\|\\_{L^2(T)}^2 \\lesssim \\| {\\bm{\\varepsilon}\\_h}- \\varepsilon(v) \\|\\_{L^2(T)}\n h\\_T^{-2} \\|{\\operatorname{rot}}^2 {\\bm{\\varepsilon}\\_h}\\|\\_{L^2(T)}.$$ \n This concludes the proof.\nThe localization of the first edge residual involves the piecewise quadratic edge-bubble function *b**E* with support *T*+ ∪ *T*− for an interior edge *E* = ∂*T*+ ∩ ∂*T*− shared by the two triangles *T*+ and *T*− with edge-patch *ω**E* : = (*T*+ ∪ *T*−). With an appropriate scaling *b**E*∣*T* = 4*λ*1*λ*2 for the two barycentric coordinates *λ*1, *λ*2 on *T* ∈ {*T*+, *T*−} associated with the two vertices of *E*. Then *b**E* ∈ *W*1, ∞(*ω**E*) and *b**E*2 ∈ *W*1, ∞(*ω**E*) satisfy 0 ≤ *b**E*2 ≤ *b**E* ≤ 1 and $|b\\_E|\\_{H^1(E)} \\lesssim h\\_E^{-1}$ etc.\nThe remaining technical detail is an extension of functions on the edge *E* to *ω**E*. Throughout this sections those functions are polynomials and given *ρ**E* ∈ *P**m*(*E*), their coefficients (in some fixed basis) already define an algebraic object that is a natural extension $\\rho\\in P\\_m(\\hat E)$ along the straight line $\\hat E:=\\operatorname{mid}(E)+{\\mathbb{R}}\\, \\tau\\_E$ that extends *E* with midpoint mid(*E*) and tangential unit vector *τ**E*. This and \n*P**E*(*ρ**E*)(*x*) : = *ρ*(*τ**E* ⋅ (*x* − mid(*E*))) for all *x* ∈ R2 \n define a linear extension operator *P**E* : *P**m*(*E*) → *C*∞(R2) with *P**E*(*ρ**E*) = *ρ**E* on *E* for any *ρ**E* ∈ *P**m*(*E*), which is constant in the normal direction, ∇*P**E*(*ρ**E*) ⋅ *ν**E* ≡ 0. This design is different from that in.\n[efficiency of first interior edge residual][lemma:eff:2] Any *v* ∈ *H*1(*ω**E*; R2), *E* ∈ E(Ω), satisfies \n$$h\\_E^{1/2} \\|\\tau\\_E\\cdot [{\\bm{\\varepsilon}\\_h}]\\_E\\tau\\_E\\|\\_{L^2(E)} \\lesssim \\| {\\bm{\\varepsilon}\\_h}- \\varepsilon(v) \\|\\_{L^2(\\omega\\_E)}.$$ \nSince $\\tau\\_E\\cdot [{\\bm{\\varepsilon}\\_h}]\\_E\\tau\\_E\\in P\\_{k+2}(E)$ is a polynomial, the above extension $P\\_E(\\tau\\_E\\cdot [{\\bm{\\varepsilon}\\_h}]\\_E\\tau\\_E)$ and the function *b* ∈ *W*02, ∞(*ω**E*) with\n \n*b*(*x*) : = *b**E*2(*x*) *ν**E* ⋅ (*x* − mid(*E*)) for all *x* ∈ R2 \ndefine some function $\\phi:= b\\, P\\_E(\\tau\\_E\\cdot [{\\bm{\\varepsilon}\\_h}]\\_E\\tau\\_E)$. Since *b* = 0 and ∇*b**E* ⋅ *ν**E* = *b**E*2 along *E*, the test function *ϕ* ∈ *H*02(*ω**E*) ⊂ *H*02(Ω) leads in Lemma [lemma:representation] to \n$$(\\tau\\_E\\cdot[{\\bm{\\varepsilon}\\_h}]\\_E\\tau\\_E, \\partial\\_{\\nu\\_E} \\phi)\\_{L^2(E)}=\n ({\\bm{\\varepsilon}\\_h}, {\\operatorname{Curl}}^2 \\phi)\\_{L^2(\\omega\\_E)} - ({\\operatorname{rot}}\\_{NC}^2{\\bm{\\varepsilon}\\_h},\\phi)\\_{L^2(\\omega\\_E)}.$$ \n Since $\\partial\\_{\\nu\\_E} \\phi=b\\_E^2 \\, \\tau\\_E\\cdot [{\\bm{\\varepsilon}\\_h}]\\_E\\tau\\_E$ on *E* and *ɛ*(*v*) ⊥ Curl2*ϕ*, an inverse estimate shows \n$$\\| \\tau\\_E\\cdot[{\\bm{\\varepsilon}\\_h}]\\_E\\tau\\_E\\|^2\\_{L^2(E)}\n \\lesssim ({\\bm{\\varepsilon}\\_h}-\\varepsilon(v), {\\operatorname{Curl}}^2 \\phi)\\_{L^2(\\omega\\_E)} - ({\\operatorname{rot}}\\_{NC}^2{\\bm{\\varepsilon}\\_h},\\phi)\\_{L^2(\\omega\\_E)}.$$ \n At the heart of the bubble-function methodology are inverse and trace inequalities that allow for appropriate scaling properties under the overall assumption of shape-regularity. In the present case, one power of *h**E* ≈ *h**T*± is hidden in the function *b* and\n \n$$\\label{scalinglawsinlemma:eff:2}\n h\\_E^{1/2}\\, | \\phi |\\_{H^2(\\omega\\_E)} \n+ h\\_E^{-3/2} \\| \\phi \\|\\_{L^2(\\omega\\_E)} \\lesssim \\| \\tau\\_E\\cdot [{\\bm{\\varepsilon}\\_h}]\\_E\\tau\\_E \\|\\_{L^2(E)}.$$ \nThe combination with the previous estimate results in \n$$\\| \\tau\\_E\\cdot[{\\bm{\\varepsilon}\\_h}]\\_E\\tau\\_E\\|^2\\_{L^2(E)}\n \\hspace{-1mm} \\lesssim \\hspace{-1mm} \\| \\tau\\_E\\cdot [{\\bm{\\varepsilon}\\_h}]\\_E\\tau\\_E \\|\\_{L^2(E)}\n \\hspace{-1mm} \n \\left( \\hspace{-1mm} h\\_E^{-1/2} \\|{\\bm{\\varepsilon}\\_h}-\\varepsilon(v)\\|\\_{L^2(\\omega\\_E)} \n \\hspace{-1mm} + \\hspace{-1mm} \n h\\_E^{3/2} \\| {\\operatorname{rot}}\\_{NC}^2{\\bm{\\varepsilon}\\_h}\\|\\_{L^2(\\omega\\_E)}\\hspace{-1mm}\\right) \\hspace{-1mm}.$$ \n This and Lemma [lemma:eff:1] conclude the proof.\nFor any boundary edge *E* ∈ E(Ω), the edge-bubble function *b**E* = 4*λ*1*λ*2 ∈ *W*1, ∞(*ω**E*) for the two barycentric coordinates *λ*1, *λ*2 associated with the two vertices of *E* and *b**E* vanishes on the remaining sides ∂*ω**E* \\ *E* of the aligned triangle $\\overline{\\omega\\_E}$. The Dirichlet data *u**D* allows for some polynomial approximation Π*m*, *E**u**D* ∈ *P**m*(*E*) of a maximal degree bounded by *m* ≥ *k* + 2; recall the definition of osc*I*(*u**D*, *E*) from.\n[efficiency of first boundary edge residual][lemma:eff:4] Any function *v* ∈ *H*1(*ω**E*; R2) with *v*∣*E* = *u**D*∣*E* along *E* ∈ E(Γ*D*) satisfies \n$$h\\_E^{1/2} \\|\\tau\\_E\\cdot ( {\\bm{\\varepsilon}\\_h}\\tau\\_E - \\partial u\\_D/\\partial s)\\|\\_{L^2(E)}\n\\lesssim \\| {\\bm{\\varepsilon}\\_h}- \\varepsilon(v) \\|\\_{L^2(\\omega\\_E)} + {\\operatorname{osc}}\\_I(u\\_D,E).$$ \nSince $\\tau\\_E\\cdot {\\bm{\\varepsilon}\\_h}\\tau\\_E$ is a polynomial of degree at most *k* + 2 ≤ *m* along the exterior edge *E*, the residual $\\tau\\_E\\cdot ( {\\bm{\\varepsilon}\\_h}\\tau\\_E - \\partial u\\_D/\\partial s)$ is well approximated by its *L*2 projection $\\rho\\_E:= (\\tau\\_E\\cdot ( {\\bm{\\varepsilon}\\_h}\\tau\\_E - \\Pi\\_{m,E} \\partial u\\_D/\\partial s))$ onto *P**m*(*E*). The Pythagoras theorem based on the *L*2 orthogonality reads \n$$h\\_E \\|\\tau\\_E\\cdot ( {\\bm{\\varepsilon}\\_h}\\tau\\_E - \\partial u\\_D/\\partial s)\\|\\_{L^2(E)}^2=\nh\\_E \\| \\rho\\_E\\|\\_{L^2(E)}^2+{\\operatorname{osc}}^2\\_I(u\\_D,E)$$ \n and it remains to bound *h**E*1/2∥*ρ**E*∥*L*2(*E*) by the right-hand side of the claimed inequality. The extension *P**E**ρ**E* ∈ *C*∞(R2) and the function *b* from lead to an admissible test function *ϕ* : = *b**P**E**ρ**E* ∈ *W*01, ∞(*ω**E*). Two successive integration by parts as in Lemma [lemma:representation] show \n(*ɛ*(*v*), Curl2*ϕ*)*L*2(*ω**E*) = (∂*u**D*/∂*s*, *τ**E*(*ν**E* ⋅ ∇*ϕ*))*L*2(*E*). \n This and Lemma [lemma:representation] lead to\n \n$$(\\tau\\_E\\cdot( {\\bm{\\varepsilon}\\_h}\\tau\\_E- \\frac{ \\partial u\\_D}{\\partial s}), \\frac{\\partial \\phi}{\\partial \\nu\\_E} )\\_{L^2(E)}=\n ({\\bm{\\varepsilon}\\_h}-\\varepsilon(v), {\\operatorname{Curl}}^2 \\phi)\\_{L^2(\\omega\\_E)} - ({\\operatorname{rot}}\\_{NC}^2{\\bm{\\varepsilon}\\_h},\\phi)\\_{L^2(\\omega\\_E)}.$$ \nSince ∂*ν**E**ϕ* = *b**E*2*ρ**E* along *E* and *ρ**E* is the *L*2 projection of $\\tau\\_E\\cdot( {\\bm{\\varepsilon}\\_h}\\tau\\_E- \\partial u\\_D/\\partial s)$, the left-hand side equals ∥*b**E**ρ**E*∥*L*2(*E*)2 − ((1 − Π*m*, *E*)∂*u**D*/∂*s*, *b**E*2*ρ**E*)*L*2(*E*). The scaling argument which leads to shows that the left-hand side of is $\\lesssim\\| \\rho\\_E\\|\\_{L^2(E)}$. The combination with the previously displayed identity leads to \n$$\\| \\rho\\_E\\|\\_{L^2(E)}^2\\lesssim \\| \\rho\\_E\\|\\_{L^2(E)} \\hspace{-1mm} \n \\left( \\hspace{-1mm} h\\_E^{-1/2} \\|{\\bm{\\varepsilon}\\_h}-\\varepsilon(v)\\|\\_{L^2(\\omega\\_E)} \n \\hspace{-1mm} + \\hspace{-1mm} h\\_E^{3/2} \\| {\\operatorname{rot}}\\_{NC}^2{\\bm{\\varepsilon}\\_h}\\|\\_{L^2(\\omega\\_E)}\n \\hspace{-1mm} + \\hspace{-1mm}h\\_E^{-1/2}\\hspace{-1mm} {\\operatorname{osc}}\\_I(E,u\\_D) \\hspace{-1mm} \\right) \\hspace{-1mm}.$$ \n This and Lemma [lemma:eff:1] conclude the proof.\nThe edge-bubble functions for the second edge residuals are defined slightly differently to ensure some vanishing normal derivative.\n[efficiency of second interior edge residual][lemma:eff:3] Any *v* ∈ *H*1(*ω**E*; R2), *E* ∈ E(Ω), satisfies \n$$h\\_E^{3/2} \\|\\tau\\_E\\cdot( [{\\operatorname{rot}}\\_{NC} {\\bm{\\varepsilon}\\_h}]\\_E - \\partial [{\\bm{\\varepsilon}\\_h}]\\_E/\\partial s\\, \\nu\\_E)\\|\\_{L^2(E)} \n \\lesssim \\| {\\bm{\\varepsilon}\\_h}- \\varepsilon(v) \\|\\_{L^2(\\omega\\_E)}.$$ \nThere are many ways to define an edge-bubble function for this situation and one may first select a maximal open ball *B*(*x**E*, 2*r**E*) ⊂ *ω**E* around a point *x**E* ∈ *E* with maximal radius 2*r**E*, which is entirely included in *ω**E*. The characteristic function *χ**B*(*x**E*, *r**E*) of the smaller ball *B*(*x**E*, *r**E*) may be regularized with a standard mollification *η**r**E* to define the smooth function *b* : = *χ**B*(*x**E*, *r**E*) \\* *η**r**E* ∈ *C**c*∞(Ω*E*) with values in [0, 1] and with ∇*b* ⋅ *ν**E* = 0 along *E*. The polynomial $\\rho\\_E:= \\tau\\_E\\cdot( [{\\operatorname{rot}}\\_{NC}{\\bm{\\varepsilon}\\_h}]\\_E - \\partial [{\\bm{\\varepsilon}\\_h}]\\_E/\\partial s \\, \\nu\\_E)$ and its extension *P**E**ρ**E* define the test function *ϕ* : = *b**P**E**ρ**E* ∈ *C*0∞(*ω**E*) in Lemma [lemma:representation]. The representation formula and (*ɛ*(*v*), Curl2*ϕ*)*L*2(*ω**E*) = 0 lead to \n$$\\| b^{1/2} \\rho\\_E\\|\\_{L^2(E)}^2\n= (\\varepsilon(v)-{\\bm{\\varepsilon}\\_h},{\\operatorname{Curl}}^2\\phi)\\_{L^2(\\omega\\_E)} + ({\\operatorname{rot}}\\_{NC}^2{\\bm{\\varepsilon}\\_h},\\phi)\\_{L^2(\\omega\\_E)}.$$ \n The inverse inequality ∥*ρ**E*∥*L*2(*E*)∥ ≤ ∥*b*1/2*ρ**E*∥*L*2(*E*), Cauchy inequalities, and the right scaling properties of *ϕ* lead to \n$$\\|\\rho\\_E\\|\\_{L^2(E)}^2\\lesssim \\|\\rho\\_E\\|\\_{L^2(E)}\\left( \nh\\_E^{-3/2} \\|{\\bm{\\varepsilon}\\_h}-\\varepsilon(v) \\|\\_{L^2(\\omega\\_E)} + h\\_E^{1/2}\\| {\\operatorname{rot}}\\_{NC}^2{\\bm{\\varepsilon}\\_h}\\| \\_{L^2(\\omega\\_E)} \\right).$$ \n This and Lemma [lemma:eff:1] conclude the proof.\nThe efficiency of the last edge contribution involves the second Dirichlet data oscillation osc*I**I*(*u**D*, *E*) from.\n[efficiency of second boundary edge residual][lemma:eff:5] Any function *v* ∈ *H*1(*ω**E*; R2) with *v*∣*E* = *u**D*∣*E* along *E* ∈ E(Γ*D*) satisfies \n$$h\\_E^{3/2} \\|\\tau\\_E\\cdot {\\operatorname{rot}}{\\bm{\\varepsilon}\\_h}- \\nu\\_E\\cdot (\\frac{ \\partial {\\bm{\\varepsilon}\\_h}\\tau\\_E }{\\partial s }\n+\\frac{ \\partial^2 u\\_D}{ \\partial s^2}) \\|\\_{L^2(E)}\n\\lesssim \\| {\\bm{\\varepsilon}\\_h}- \\varepsilon(v) \\|\\_{L^2(\\omega\\_E)} + {\\operatorname{osc}}\\_{II}(u\\_D,E).$$ \nSelect a maximal open ball *B*(*x**E*, 2*r**E*) ∩ Ω ⊂ *ω**E* around a point *x**E* ∈ *E* with maximal radius 2*r**E* such that *B*(*x**E*, 2*r**E*) ∩ *ω**E* is a half ball. The regularization *b* : = *χ**B*(*x**E*, *r**E*) \\* *η**r**E* ∈ *C**c*∞(R2) of the characteristic function *χ**B*(*x**E*, *r**E*) attains values in [0, 1] and a positive integral mean *h**E*− 1∫*E**b* *d**s* ≈ 1 along *E* (depending only on the shape regularity of T); *b* vanishes on ∂*ω**E* \\ *E* and its normal derivative ∇*b* ⋅ *ν* = 0 vanishes along the entire boundary ∂*ω**E*.\nThe Pythagoras theorem ∥*ρ*∥*L*2(*E*)2 = ∥*ρ**E*∥*L*2(*E*)2 + *h**E*− 3osc*I**I*2(*u**D*, *E*) for the residual $\\rho:=\\tau\\_E\\cdot {\\operatorname{rot}}{\\bm{\\varepsilon}\\_h}- \\nu\\_E\\cdot (\\frac{ \\partial {\\bm{\\varepsilon}\\_h}\\tau\\_E }{\\partial s }\n +\\frac{ \\partial^2 u\\_D}{ \\partial s^2})$ and its *L*2 projection *ρ**E* : = Π*m*, *E**ρ* onto *P**m*(*E*) reduces the proof to the estimation of ∥*ρ**E*∥*L*2(*E*). The normal derivative of $\\phi:= b\\, P\\_E \\rho\\_E\\in C^\\infty(\\overline{\\omega\\_E})$ vanishes along the boundary ∂*ω**E* and Lemma [lemma:representation] shows \n$$({\\operatorname{rot}}{\\bm{\\varepsilon}\\_h}-\\frac{ \\partial {\\bm{\\varepsilon}\\_h}\\nu\\_E}{\\partial s}, b \\rho\\_E \\tau\\_E)\\_{L^2(E)}\n= ({\\operatorname{rot}}\\_{NC}^2{\\bm{\\varepsilon}\\_h},\\phi)\\_{L^2(\\omega\\_E)}-({\\bm{\\varepsilon}\\_h}, {\\operatorname{Curl}}^2 \\phi)\\_{L^2(\\omega\\_E)}.$$ \n The arguments in Lemma [lemma:bd] show (∂2*u**D*/∂*s*2, *b**ρ**E* *ν**E*)*L*2(*E*) = (*ɛ*(*v*), Curl2*ϕ*)*L*2(*ω**E*). The combination of the two identities leads to a formula for (*ρ*, *b**ρ**E*)*L*2(*E*). Since *ρ* − *ρ**E* is controlled in osc*I**I*2(*u**D*, *E*), this and an inverse inequality in the beginning result in\n \n$$\\begin{aligned}\n&\\|\\rho\\_E\\|\\_{L^2(E)}^2\\lesssim (b\\rho\\_E,\\rho\\_E) \\_{L^2(E)}\n=(\\rho, b\\rho\\_E) \\_{L^2(E)}- (\\rho-\\rho\\_E, b\\rho\\_E) \\_{L^2(E)} \\\\\n&\\; \\lesssim ({\\operatorname{rot}}\\_{NC}^2{\\bm{\\varepsilon}\\_h},\\phi)\\_{L^2(\\omega\\_E)}\n-({\\bm{\\varepsilon}\\_h}-\\varepsilon(v), {\\operatorname{Curl}}^2 \\phi)\\_{L^2(\\omega\\_E)} +\\|\\rho\\_E\\|\\_{L^2(E) }h\\_E^{-3/2} {\\operatorname{osc}}\\_{II}(u\\_D,E).\\end{aligned}$$ \nThe scaling properties of *ϕ* and its derivatives are as in the proof of the previous lemma. With Lemma [lemma:eff:1] in the end, this concludes the proof.\nNumerical examples\n==================\nThis section is devoted to numerical experiments for four different domains to demonstrate robustness in the reliability and efficiency of the a posteriori error estimator *η*(Tℓ, *σ*ℓ). The implementation follows for *k* = 1 with Lamé parameters *λ* and *μ* from *λ* = *E**ν*/((1 + *ν*)(1 − 2*ν*)) and *μ* = *E*/(2(1 + *ν*)) for a Young’s modulus *E* = 105 and various Poisson ratios *ν* = 0.3 and *ν* = 0.4999.\nAcademic example\n----------------\n\n[fig:Convergence1]\nThe model problem on the unit square Ω = (0, 1)2 with homogeneous Dirichlet boundary conditions and the right-hand side *f* = (*f*1, *f*2), \n*f*1(*x*, *y*) = − *f*2(*y*, *x*) = − 2*μ**π*3cos(*π**y*)sin(*π**y*)(2cos(2*π**x*) − 1) for (*x*, *y*) ∈ Ω, \n allows the smooth exact solution \n*u*(*x*, *y*) = *π* sin(*π**x*)sin(*π**y*) (cos(*π**y*)sin(*π**x*), − cos(*π**x*)sin(*π**y*)) for (*x*, *y*) ∈ Ω. \n Note that *f* depends only on the Lamé parameter *μ* and not on the critical Lamé parameter *λ*. For uniform mesh refinement, Figure [fig:Convergence1] displays the robust third-order convergence of the a posteriori error estimator *η*(Tℓ, *σ*ℓ) as well as the Arnold-Winther finite element stress error. The convergence is robust in the Poisson ratio *ν* → 1/2 and the a posteriori error estimator proves to be reliable and efficient. In this example, the oscillations osc(*f*, Tℓ) dominate the a posteriori error estimator.\nThis typical observation motivates numerical examples with *f* ≡ 0 in the sequel.\nCircular inclusion\n------------------\n[scale = 4]\n(0.25,0) – (1,0); (1,0) – (1,1); (1,1) – (0,1); (0,0.25) – (0,1);\n(0.25,0) arc (0:90:0.25);\nin 0,...,9 (1,0.1+0.05) – ++(0.1,0); \nin 3,...,9 (-0.05,0.1+0.05) circle (0.035); \nin 3,...,9 (0.1+0.05,-0.05) circle (0.035); \n(0.1,0.1) node Γ*D*;\n(1.2,0.5) node Γ*N*;\n(-0.15,0) – (-0.15,0.25); (-0.2,0.1) node[rotate=90] 25;\n(-0.15,0.25) – (-0.15,1); (-0.2,0.6) node[rotate=90] 75;\n(0,-0.15) – (0.25,-0.15); (0.1,-0.2) node 25;\n(0.25,-0.15) – (1,-0.15); (0.6,-0.2) node 75;\nin 1,...,9 (9:0.25) – (9:0.2); \n[fig:ex:geom:CircularIncusion]\n\n[fig:Convergence2]\nThe second benchmark example from the literature models a rigid circular inclusion in an infinite plate for the domain Ω with rather mixed boundary conditions indicated with mechanical symbols in Figure [fig:ex:geom:CircularIncusion]. The exact solution to the model problem reads (with polar coordinates (*r*, *ϕ*) and *κ* = 3 − 4*ν*, *γ* = 2*ν* − 1, *a* = 1/4)\n \n$$\\begin{aligned}\n u\\_r &= \\frac{1}{8\\mu r}\\left( (\\kappa-1)r^2 + 2\\gamma a^2 + \n \\left( 2r^2 -\\frac{2(\\kappa+1)a^2}{\\kappa} + \\frac{2 a^4}{\\kappa r^2}\\right) \\cos(2\\phi)\\right),\\\\\n u\\_\\phi &= -\\frac{1}{8\\mu r}\\left( 2r^2-\\frac{2(\\kappa-1)a^2}{\\kappa} - \\frac{2a^4}{\\kappa r^2}\\right)\\sin(2\\phi).\\end{aligned}$$ \nThe approximation of the circular inclusion through a polygon is rather critical for the higher-order Arnold–Winther MFEM. In the absence of an implementation of parametric boundaries, adaptive mesh refinement is necessary for higher improvements. The adaptive algorithm of this section is the same for all examples and acts on polygons; in particular, it does not monitor the curved boundary, but whenever some edge at the curved part Γ*D* is refined in this example, the midpoint is a new node and projected onto Γ*D*. The convergence history plot in Figure [fig:Convergence2] shows a reduced convergence for uniform refinement, while adaptive refinement (of the circular boundary) leads to optimal third-order convergence.\nL-shaped benchmark\n------------------\n[scale=1.25] (0,0) coordinate (A); (-1,-1) coordinate (B); (0,-2) coordinate (C); (2,0) coordinate (D); (0,2) coordinate (E); (-1,1) coordinate (F);\n(A) – (B) – (C) – (D) – (E) – (F) – (A);\n(0,0) – (2.5,0); (0,0) – (0,2.5);\n(2.5,-0.1) node *x*; (-0.1,2.5) node *y*; (1.3,1.3) node Γ*D*;\n(0,0) – (0.5,0.5); (0.25,0.5) node *r*;\n(0.5,0) arc (0:45:0.5); (0.55,0.25) node *ϕ*;\n(-1,-2.25) – (0,-2.25); (0,-2.25) – (2,-2.25);\n(-1.25,-2) – (-1.25,2); (-0.5,-2.4 ) node 1; (1,-2.4) node 2; (-1.4,0) node[rotate=90] 4;\n(-0.5,0) node Γ*N*;\nin 0,...,10 (-1+0.1,-1-0.1) – ++(0,-0.15); (-1+0.1,1+0.1) – ++(0,0.15); \nin 0,...,20 (0.1,-2+0.1) – ++(0,-0.15); (0.1,2-0.1) – ++(0,0.15); \n[fig:geom:Lshape]\n\n[fig:Convergence3]\nConsider the rotated L-shaped domain with Dirichlet and Neumann boundary depicted in Figure [fig:geom:Lshape]. The exact solution reads in polar coordinates\n \n$$\\begin{aligned}\n u\\_r(r,\\phi) &= \\frac{r^{\\alpha}}{2\\mu} \\left( -(\\alpha+1)\\cos((\\alpha+1)\\phi) + (C\\_2-\\alpha-1)C\\_1\\cos((\\alpha-1)\\phi)\\right),\\\\\n u\\_\\phi(r,\\phi) &= \\frac{r^{\\alpha}}{2\\mu} \\left( (\\alpha+1)\\sin((\\alpha+1)\\phi) + (C\\_2+\\alpha-1)C\\_1\\sin((\\alpha-1)\\phi)\\right).\\end{aligned}$$ \nThe constants are *C*1 : = − cos((*α* + 1)*ω*)/cos((*α* − 1)*ω*) and *C*2 : = 2(*λ* + 2*μ*)/(*λ* + *μ*), where *α* = 0.544483736782 is the first root of *α*sin(2*ω*) + sin(2*ω**α*) = 0 for *ω* = 3*π*/4. The volume force *f* ≡ 0 and the Neumann boundary data *g* ≡ 0 vanish, and the Dirichlet boundary conditions *u**D* are extracted from the exact solution.\nFigure [fig:Convergence3] shows suboptimal convergence O(*N*ℓ− 0.27), namely an expected rate *α* in terms of the maximal mesh-size, for uniform and fourth-order *L*2 stress convergence for adaptive mesh-refinement.\nDespite the singular solution, the adaptive algorithm recovers the higher convergence of Theorem [supperconvergence] as in.\nCook membrane problem\n---------------------\n[scale = 0.08]\n(0,0) coordinate (A); (0,44) coordinate (B); (48,60) coordinate (C); (48,44) coordinate (D);\n(D) – (A) – (B) – (C); (B) – (D);\nin 1,...,21 (0,2) – ++(-3,-3); (48,44) – (48,52); (48,52) – (48,60); (48,60) – (48,65);\n(-6,22) node[fill=white] Γ*D* ; (54,53) node Γ*N*;\n(0,-5) – (48,-5); (24,-8) node 48;\n(-10,0) – (-10,44); (-10,44) – (-10,60); (-15,22) node[rotate=90] 44; (-15,52) node[rotate=90] 16;\n(A) node[right] A; (B) node[left] D; (C) node[right] C; (D) node[below] B;\n[fig:geom:Cooks]\n\n[fig:Convergence4]\nOne of the more popular benchmarks in computational mechanics is the tapered panel Ω with the vertices *A*, *B*, *C*, *D* of Figure [fig:geom:Cooks] clamped on the left side Γ*D* = conv(*D*, *A*) (with *u**D* ≡ 0) under no volume force (*f* ≡ 0) but applied surface tractions *g* = (0, 1) along conv(*B*, *C*) and traction free on the remaining parts conv(*A*, *B*) and conv(*C*, *D*) along the Neumann boundary.\nThis example is a particular difficult one for the Arnold–Winther MFEM because of the incompatible Neumann boundary conditions on the right corners. That means, although *g* is piecewise constant, *g* does not belong to *G*(T) for any triangulation. In the two Neumann corner vertices *B* and *C* we therefore strongly impose the values *σ*ℓ(*B*) = (0.2491, 0.7283; 0.7283, 0.6676) and *σ*ℓ(*C*) = (3/20, 11/20; 11/20, 11/60) for the design of *g*ℓ ∈ *G*(Tℓ).\nSince the exact solution is unknown, the error approximation rests on a reference solution $\\tilde \\sigma$ computed as *P*5(T) displacement approximation on the uniform refinement of the finest adapted triangulation.\nThe large pre-asymptotic range of the convergence history plot in Figure [fig:Convergence4] illustrates the difficulties of the Arnold-Winther finite element method in case of incompatible Neumann boundary conditions according to its nodal degrees of freedom. Once the resulting and dominating boundary oscillations (caused by the necessary choice of discrete compatible Neumann conditions in *G*(Tℓ)) osc(*g* − *g*ℓ, Eℓ(Γ*N*)) are resolved through adaptive mesh-refining, even the fourth-order *L*2 stress convergence is visible in a long asymptotic range in (the approximated error and) the equivalent error estimator.\nThis example underlines that adaptive mesh-refining is unavoidable in computational mechanics with optimal rates and a large saving in computational time and memory compared to naive uniform mesh-refining.\nAcknowledgements\n================\nThe work has been written, while the three authors enjoyed the hospitality of the Hausdorff Research Institute of Mathematics in Bonn, Germany, during the Hausdorff Trimester Program *Multiscale Problems: Algorithms, Numerical Analysis and Computation*. The research of the first author (CC) has been supported by the Deutsche Forschungsgemeinschaft in the Priority Program 1748 ‘*Reliable simulation techniques in solid mechanics. Development of non-standard discretization methods, mechanical and mathematical analysis*’ under the project ‘*Foundation and application of generalized mixed FEM towards nonlinear problems in solid mechanics*’ (CA 151/22-1). The second author (DG) has been supported by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173. The third author (JG) has been funded by the Austrian Science Fund (FWF) through the project P 29197-N32.\n0\nD. N. Arnold, F. Brezzi, and J. Douglas. : A new mixed finite element for plane elasticity., 1:347–367, 1984.\nD. N. Arnold, R. S. Falk, and R. Winther. Finite element exterior calculus, homological techniques, and applications., 15:1–155, 2006.\nA. Alonso. Error estimators for a mixed method., 74(4):385–395, 1996.\nD. N. Arnold. Differential complexes and numerical stability. In *Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002)*, pages 137–157. Higher Ed. Press, Beijing, 2002.\nD. N. Arnold and R. Winther. Mixed finite elements for elasticity., 92(3):401–419, 2002.\nD. Boffi, F. Brezzi, and M. Fortin., volume 44 of *Springer Series in Computational Mathematics*. Springer, Heidelberg, 2013.\nD. Braess.. Cambridge University Press, Cambridge, third edition, 2007.\nS. C. Brenner and L. R. Scott., volume 15 of *Texts in Applied Mathematics*. Springer, New York, third edition, 2008.\nC. Carstensen. A posteriori error estimate for the mixed finite element method., 66(218):465–476, 1997.\nC. Carstensen. A unifying theory of a posteriori finite element error control., 100(4):617–637, 2005.\nC. Carstensen and G. Dolzmann. A posteriori error estimates for mixed FEM in elasticity., 81(2):187–209, 1998.\nC. Carstensen, M. Eigel, and J. Gedicke. Computational competition of symmetric mixed FEM in linear elasticity., 200(41-44):2903–2915, 2011.\nC. Carstensen, M. Feischl, M. Page, and D. Praetorius. Axioms of adaptivity., 67(6):1195–1253, 2014.\nC. Carstensen and J. Gedicke. Robust residual-based a posteriori Arnold-Winther mixed finite element analysis in elasticity., 300:245–264, 2016.\nC. Carstensen, D. Günther, J. Reininghaus, and J. Thiele. The Arnold-Winther mixed FEM in linear elasticity. Part I: Implementation and numerical verification., 197:3014–3023, 2008.\nC. Carstensen, D. Gallistl, and M. Schedensack. best approximation of the elastic stress in the Arnold-Winther FEM., 36(3):1096–1119, 2016.\nC. Carstensen and J. Hu. A unifying theory of a posteriori error control for nonconforming finite element methods., 100(3):473–502, 2007.\nP. G. Ciarlet., volume 4 of *Studies in Mathematics and its Applications*. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978.\nC. Carstensen, D. Peterseim, and A. Schröder. The norm of a discretized gradient in *H*(div)\\* for a posteriori finite element error analysis., 132:519–539, 2016.\nC. Carstensen and H. Rabus. Axioms of adaptivity for separate marking., arXiv:1606.02165, 2016.\nV. Girault and L. R. Scott. Hermite interpolation of nonsmooth functions preserving boundary conditions., 71(239):1043–1074, 2002.\nJ. Huang, X. Huang, and Y. Xu. Convergence of an adaptive mixed finite element method for Kirchhoff plate bending problems., 49(2):574–607, 2011.\nR. Kouhia and R. Stenberg. A linear nonconforming finite element method for nearly incompressible elasticity and Stokes flow., 124(3):195–212, 1995.\nR. Verfürth.. Wiley and Teubner, 1996.\nAppendix: Fourth-order convergence of the stress in *L*2\n========================================================\nThis appendix explains a high-order convergence phenomenon observed in some numerical benchmark examples for the lowest-order Arnold-Winther method. Adopt the notation from this paper for *k* = 1 and let *σ* solve and let *σ**h* ∈ *A**W**k*(T) solve.\n[fourth-order convergence][supperconvergence] Suppose *f* = *f**h* ∈ *P*1(T; R2) and *g* = *g**h* ∈ *G*(T) and suppose that the stress *σ* ∈ *H*4(Ω; S). Then the *L*2 stress error satisfies (with the maximal mesh-size *h*) \n$$\\|\\sigma-\\sigma\\_h\\|\\_{L^2(\\Omega)} \\lesssim h^4 \\|\\sigma\\|\\_{H^4(\\Omega)}.$$ \nSince the stress error *σ* − *σ**h* ∈ *H*4(T; S) is divergence-free, *α* vanishes in and *σ* − *σ**h* = Curl2*β* ∈ *H*4(T; S). Since *β* ∈ *H*2(T) is piecewise in *C*2, it follows $\\beta\\in C^1(\\overline{\\Omega})$. The Arnold-Winther finite elements have nodal degrees of freedom at the vertices and hence *σ**h* is continuous at each vertex *z* ∈ N. Hence the second derivatives of $\\beta\\in C^2({\\mathcal{T}})\\cap C^1(\\overline{\\Omega})$ are continuous at each vertex *z* ∈ N. It follows that the nodal interpolation operator *I**A* associated to the Argyris finite element space *A*(T) ⊂ *C*1(Ω) ∩ *P*5(T) exists for *β* in the classical sense and is composed of the piecewise local interpolation. This defines *β**h* = *I**A**β* and the divergence-free *τ**h* : = Curl2*β**h* ∈ *A**W*(T) test function in and in. Consequently, \n∥*σ* − *σ**h*∥*L*2(Ω)2 = (*σ* − *σ**h*, Curl2(*β* − *β**h*))*L*2(Ω) ≤ ∥*σ* − *σ**h*∥*L*2(Ω) ∣*β* − *I**A**β*∣*H*2(Ω). \n This and standard local interpolation error estimates for the nodal interpolation of the quintic Argyris finite elements show \n$$\\|\\sigma-\\sigma\\_h\\|\\_{L^2(\\Omega)}\\lesssim h^4 |\\beta |\\_{H^6({\\mathcal{T}})}=h^4 |\\sigma |\\_{H^4(\\Omega)}.$$ \n (With *σ* − *σ**h* = Curl2*β* and ∣*σ**h*∣*H*4(T) = 0 for piecewise cubic *σ**h* in the last step.)\n---\n1. Humboldt-Universität zu Berlin, Berlin, Germany[↩](#fnref1)\n2. Karlsruher Institut für Technologie, Karlsruhe, Germany[↩](#fnref2)\n3. \n"},"data_id":{"kind":"string","value":"arxiv_0185565"}}},{"rowIdx":46396,"cells":{"content":{"kind":"string","value":"Two-way classical communication remarkably improves local distinguishability\n============================================================================\nWe analyze the difference in the local distinguishability among the following three restrictions; (i) Local operations and only one-way classical communications (one-way LOCC) are permitted. (ii) Local operations and two-way classical communications (two-way LOCC) are permitted. (iii) All separable operations are permitted. We obtain two main results concerning the discrimination between a given bipartite pure state and the completely mixed state with the condition that the given state should be detected perfectly. As the first result, we derive the optimal discrimination protocol for a bipartite pure state in the cases (i) and (iii). As the second result, by constructing a concrete two-way local discrimination protocol, it is proven that the case (ii) is much better than the case (i), i.e., two-way classical communication remarkably improves the local distinguishability in comparison with one-way classical communication at least for a low-dimensional bipartite pure state.\n: state discrimination, one-way LOCC, two-way LOCC, Schmidt rank, Schmidt coefficient\nIntroduction\n============\nRecently, quantum communication has been investigated among many groups as a future technology. Similar to conventional information technology, practical quantum communication technology will require distributed information processing among two or more spatially separated parties. In order to treat this problem, it is necessary to clarify what kind of information processing is possible under respective constraints for permitted operations. In the quantum case, when our quantum system consists of distinct two parties *A* and *B*, we often restrict our operations to local (quantum) operations and classical communications (LOCC) because sending quantum states over long distance is technologically more difficult than sending classical information. Even in this restriction, we can consider the following two formulations; (i) The classical communication is restricted to the direction *A* to *B* (We can similarly treat the restriction of the opposite direction.) (ii) All parties are allowed to communicate classically with each other as much as they like. The case (i) is called the one-way LOCC, and the case (ii) is called the two-way LOCC.\nSince, by definition, the two-way LOCC apparently includes the one-way LOCC, the two-way LOCC is always more powerful than the one-way LOCC in principle. However, due to the following two reasons, it is not easy to characterize the difference between the both performances. As the first reason, it is mathematically hard to rigorously evaluate the performance of a given two-way LOCC protocol, because the mathematical description of two-way LOCC is too complicated. As the second reason, for several simple tasks, the performance of the two-way LOCC was actually shown to be the same as the performance of the one-way LOCC. In fact, there are several settings that has no difference between one-way LOCC and two-way LOCC, e.g., LOCC convertibility of bipartite pure state, Stein’s lemma bound in the simple asymptotic hypothesis testing of the *n*-tensor product of identical states.\nOn the other hand, in several settings, the two-way LOCC is strictly powerful than the one-way LOCC. For example, the distillable entanglement (the amount of maximally entangled states which can be derived from a given state by LOCC) with two-way classical communication is proven to be greater than that with one-way classical communication. Also, this type comparison also has been done by several papers only on the discrimination among orthogonal states. Although several researchers treated this problem, they did not treated the discrimination among non-orthogonal states. In this paper, we compare the both performances quantitatively on the “*local discrimination*” among states that are not necessarily orthogonal, whose purpose is discriminating given states by only LOCC with the *single* copy. In fact, general discrimination problem is closely related to sending classical information via quantum channel and quantum algorithm.\nIn order to quantify the difference of the two-way LOCC and the one-way LOCC, in this paper, we concentrate on a simple setting: the local discrimination of the first state *ρ* on a bipartite system H from the second state *ρ̃* under the condition where the first state *ρ* should be detected perfectly. When the both states *ρ* and *ρ̃* are pure, there is no difference between one-way LOCC and two-way LOCC because any global discrimination protocol can be simulated by one-way LOCC. Surprisingly, as our result, we found that there usually exists non-negligible difference between two restrictions when the second *ρ̃* is the completely mixed state $\\rho\\_{\\rm mix}:=I/\\dim {\\mathcal{H}}$. At the first glance, this setting seems specific, however, due to the following six reasons, it is closely related to several research topics. First, this type analysis produces a bound of the number of perfectly locally distinguishable states. Second, as is explained later, there is a relation between the performance of local distinguishability and amount of entanglement in the case of pure states. Third, this kind of distinguishability is often treated in quantum complexity as Triviality of Coset State. Fourth, when the second state *ρ̃* is close to the completely mixed state $\\rho\\_{\\rm mix}$, we obtain a similar conclusion because the power of our test is continuous concerning the second state. Fifth, in the community of classical statistics, the problem of discriminating the given two distributions is widely accepted as the fundamental problem of hypothesis testing because general hypothesis testing problem can be treated by using this type problem. Sixth, as was mentioned in the preceding papers, hypothesis testing with two candidates states is closely related to quantum channel coding. Hence, it is suitable to treat this kind of local discrimination problem.\nIn order to analyze this problem in the respective settings, we introduce the minimum error probabilities to detect the complete mixed state *β*→(*ρ*), *β*↔(*ρ*), and $\\beta\\_{\\rm sep }(\\rho )$ by the one-way LOCC, the two-way LOCC, and the separable operations, respectively. Indeed, these functions are considered as appropriate measures of the local distinguishability because they give not only the minimum error probability of the above problem, but also the upper bound of the size of locally distinguishable sets in general perfect local discrimination problems. Under this formulation, we first analyze the local distinguishability by means of one-way LOCC and separable operations, and derive the optimal discrimination protocol with one-way LOCC and separable operations; we should note that the minimum error probability $\\beta\\_{\\rm sep }(\\rho )$ with separable operations gives a lower bound for the minimum error probability *β*↔(*ρ*) with two-way LOCC. After that, constructing a concrete two-way local discrimination protocol, we show that two-way classical communication remarkably improves the local distinguishability in comparison with the local discrimination by one-way classical communication at least for a low-(less than five) dimensional bipartite pure state. Indeed, since the power of our test is continuous concerning the first and the second states, our result indicates that two-way classical communication remarkably improves the local distinguishability in a wider class of the first and the second states. Moreover, as a byproduct, we extend the characterization of locally distinguishability by one-way LOCC by Cohen to a set of mixed states.\nThis paper is organized as follows: In Section [preliminary], we introduce the discrimination problem between an arbitrary given state *ρ* and a completely mixed state $\\rho\\_{\\rm mix}$ on a bipartite system H under the condition that the given state is detected perfectly. Then, we explain another meaning of *β*→(*ρ*), *β*↔(*ρ*), and $\\beta\\_{\\rm sep }(\\rho )$ from the viewpoint of general local discrimination problems. In Section [separable section chap.3], constructing the optimal separable POVM for the local discrimination, we prove that $D \\beta\\_{\\rm sep }({{\\left \\vert}\\Psi {\\right \\rangle}} )-1$ coincides with the entanglement monotone called robustness of the entanglement for a bipartite pure state, where *D* is the dimension of the bipartite Hilbert space H. In Section [bipartite], we show that the amount $D \\beta\\_{\\rightarrow } ({{\\left \\vert}\\Psi {\\right \\rangle}})$ with one-way LOCC coincides with the Schmidt rank (the rank of the reduced density matrix) of the states. Also, as a corollary, we extend Cohen’s characterization to a set of mixed states. Finally, in section [two-way], constructing a concrete three-step two-way LOCC discrimination protocol, we derive an upper bound for *β*↔(*ρ*). Calculating this upper bound analytically and also numerically, we show that $\\beta\\_{\\leftrightarrow}({{\\left \\vert}\\Psi {\\right \\rangle}})$ is strictly smaller than $\\beta\\_{\\rightarrow } ({{\\left \\vert}\\Psi {\\right \\rangle}} )$, and moreover, *β*→(*ρ*) and $\\beta\\_{\\rm sep} (\\rho)$ give almost the same value for a lower dimensional bipartite pure state; this results can be seen in **Figures 2,3,4,5,6**. As a result, we conclude that the two-way classical communication remarkably improves the local distinguishability in comparison with the one-way classical communication for a low-dimensional pure state at least in the present problem settings.\nLocal discrimination between an arbitrary state and the completely mixed state\n==============================================================================\nIn this paper, we treat the bipartite system H : = H*A* ⊗ H*B* (dimH = *D*) composed of two finite-dimensional subsystems H*A* and H*B*. In the following sections, we often focus on the case when *ρ* is pure. In such a case, we assume that the dimension *d* of H*A* is equal to that of H*B*. Note that the given pure state belongs to the composite system of the same-dimensional subsystem. Then, the dimension (*D*) of the Hilbert space H is equal to *d*2. In the composite system H, we call a positive operator *T* with 0 ≤ *T* ≤ *I* a one-way LOCC POVM element, where *I* is an identity operator on H, if the two-valued POVM {*T*, *I* − *T*} can be implemented by the one-way LOCC; we also define a two-way LOCC POVM element and a separable POVM element in the same manner by using the two-way LOCC and the separable operations in stead of the one-way LOCC, respectively. We write a set of one-way LOCC, two-way LOCC, separable POVM elements, and all (global) POVM elements as T→, T↔, $\\mathcal{T} \\_{\\rm sep}$, and $\\mathcal{T}\\_{\\rm g}$. Obviously, they satisfy the relation $\\mathcal{T}\\_{\\rightarrow} \\subset\n\\mathcal{T}\\_{\\leftrightarrow} \\subset \\mathcal{T}\\_{\\rm sep} \\subset \\mathcal{T}\\_{\\rm g}$. We can see that the condition *T* ∈ T*c* is equivalent with the condition *I* − *T* ∈ T*c*, where *c* can be either $\\rightarrow,\n\\leftrightarrow, {\\rm sep}$, or ${\\rm g}$.\nIn this paper, we discuss the comparison of the performance of the local discrimination in the case of the one-way LOCC, the two-way LOCC, and the separable operations. In order to find this difference, although there are many problem settings for the local discrimination, we especially focus on one of the simplest problem settings as follows: We consider local discrimination of an given arbitrary state *ρ* and another state *ρ̃*, and investigate how well we can detect *ρ̃* under the additional condition that we do not make any error to detect *ρ* when the second state *ρ̃* is the completely mixed state $\\rho \\_{\\rm mix} \\stackrel{\\rm def}{=} \\frac{I\\_{AB}}{D } $ (*D* = dimH); namely, by only LOCC, how well we can distinguish a given entangled state *ρ* from the white noise state $\\rho\\_{\\rm mix}$ without making any error to judge the given state is $\\rho \\_{\\rm mix}$ when the real state is *ρ*.\nOur problem can be written down rigorously as follows. We measure an unknown state chosen from two candidates {*ρ*, *ρ̃*} by the two-values POVM {*T*, *I* − *T*}, where $T \\in \\mathcal{T} \\_{\\rightarrow }, \\mathcal{T} \\_{\\leftrightarrow\n}, \\mathcal{T}\\_{\\rm sep}, \\mathrm{or} \\ \\mathcal{T}\\_{\\rm g}$; that is, if we get the result corresponding to *T*, then we decide that the unknown state is in *ρ*, and if we get the result corresponding to *I* − *T*, then we decide that the unknown state is in *ρ̃*. We consider two kinds of error probability as follows: the type 1 error probability $\\Tr\n\\rho (I-T)$, and the type 2 error probability $\\Tr \\tilde{\\rho} T$; these are common terms in the field of “*quantum hypothesis testing*”, where these two different error probabilities are treated in an asymmetric way. In this case, the type 1 error probability corresponds to the error probability that the real state is *ρ* and our decision is *ρ̃*, and the type 2 error probability corresponds to the error probability that the real state is *ρ̃* and our decision is *ρ*. Thus, our problem is to minimize the type 2 error probability $\\Tr \\tilde{\\rho} T$ under the additional condition that the type 1 error probability $\\Tr \\rho\n(I-T)$ must be 0. Thus, we focus on the following minimum of the type 2 error probability:\n \n$$\\beta \\_c (\\rho \\| \\tilde{\\rho} ) := \n\\min \\{ \\Tr (\\tilde{\\rho}\nT) | T \\in \\mathcal{T}\\_c, \\Tr \\rho T = 1 \\},$$ \nwhere *c* = → (one-way LOCC), ↔ (two-way LOCC), ${\\rm sep}$(separable operations), and ${\\rm g}$(global operations). When the both states *ρ* and *ρ̃* are pure states ∣Φ⟩ and ∣Ψ⟩, this quantity does not depend on whether $c = \\rightarrow, \\leftrightarrow, {\\rm sep}, {\\rm or} \\ {\\rm g}$, and is calculated as\n \n*β**c*(∣Φ⟩∥∣Ψ⟩) = ∣⟨Φ∣Ψ⟩∣2 \nfor *c* = → , ↔ , ${\\rm sep}$, and ${\\rm g}$. This is because any discriminating protocol between two pure bipartite states can be simulated by one-way LOCC when we focus only on the distribution of the outcome. In this paper, we focus on the minimum of the type 2 error probability in the case of $\\tilde{\\rho}=\\rho\\_{\\rm mix}$:\n \n$$\\beta \\_c (\\rho ) :=\\beta \\_c (\\rho \\| \\rho\\_{\\rm mix} )= \\frac{t\\_c (\\rho )}{D},\n\\label{normalization}$$ \nwhere *t**c*(*ρ*) is defined as\n \n$$\\label{definition of t\\_c}\nt\\_c(\\rho ) = \\min \\{ \\Tr T | T \\in \\mathcal{T}\\_c, \\Tr T \\rho = 1\n\\}.$$ \nand *D* is the dimension of the whole system H. That is, *t**c*(*ρ*) is in proportion to the minimum of the type 2 error probability *β**c*(*ρ*) of one-way LOCC, two-way LOCC, separable POVM and global POVM in the case where *c* = → , ↔ , ${\\rm sep}$, and ${\\rm g}$, respectively. Trivially,\n \n$$t\\_{{\\rm g}}(\\rho ) = {\\mathop{\\rm rank}}\\rho.$$ \nObviously, *t**c*(*ρ*) satisfies the inequality $t\\_{{\\rm g}}(\\rho ) \\le t\\_{{\\rm sep}}(\\rho ) \\le\nt\\_{\\leftrightarrow}(\\rho ) \\le t\\_{\\rightarrow}(\\rho )$; as a matter of course, *β**c*(*ρ*) also satisfies the similar inequality. Note that by normalizing *β**c*(*ρ*) as the above Eq.([normalization]), the resulting function *t**c*(*ρ*) is no more a function depending both on *ρ* and $\\rho\\_{\\rm mix }$, but a function depending only on *ρ*.\nIn quantum information community, many papers treats the Bayesian framework, in which the Bayesian prior distribution is assumed. However, in statistics community, non-Bayesian framework is more widely accepted, in which no Bayesian prior distribution is assumed. This is because it is usually quite difficult to decide the Bayesian prior distribution based on the prior knowledge. In order to resolve this difficulty, they often treat the two kinds of error probabilities in an asymmetric way in hypothesis testing without assuming prior distribution because the importance of both error are not equal in a usual case, e.g., Neyman-Pearson lemma, Stein’s lemma, Hoeffding bound. These quantum cases are treated by several papers. In this paper, according to conventional statistics framework, we focus on the error probabilities of the first and second, and minimize the second kind of error probability under the constraint for the first one.\nHere, we explain the reason why we choose the above special problem of discrimination of an arbitrary state *ρ* from the completely mixed state $\\rho\\_{\\rm mix}$, and the reason why we add the above additional condition of perfect detection of *ρ*. As we already said before, the first reason is that this additional condition makes the analysis of the problem extremely easier. Actually as we will see later in this paper, we can derive the optimal POVM of this restricted local-discrimination problem with respect to each one-way LOCC and separable operations for a bipartite pure state. As a result, we make the difference between one-way LOCC and two-way LOCC clear for our local-discrimination problem; this is our main purpose in this paper. Note that it is generally a hard problem to find an optimal protocol for a local-discrimination problem, and only in very limited situations, optimal local-discrimination protocols are known. The second reason is that we can clearly see the relationship between local distinguishability and entanglement of a state in this problem setting. In the previous paper, we showed the relationship between local distinguishability of a set of states and an average of the values of entanglement monotones for the states in terms of inequalities. However, in this paper, we will show that the minimum error probability *β**c*(*ρ*) of our problem is proportion to entanglement monotones in the case of one-way LOCC (*c* = → ) and separable operations ($c = {\\rm sep} $) at least for bipartite pure states except an unimportant constant factor. The third reason is that the minimum error probability *β**c*(*ρ*) can give a bound of local distinguishability for a more general local discrimination problem: Suppose that a set of states {*ρ**i*}*i* = 1*N**c* is perfectly locally distinguishable by one-way LOCC (*c* = → ), two-way LOCC (*c* = ↔ ), or separable ($c = \\rm sep$) POVM. From the result obtain in the the previous paper, *t**c*(*ρ**i*) (which corresponds to *d*(*ρ*)) gives an upper bound of *N**c* as,\n \n$$\\label{local dicrimination bound}\n N \\_c \\le D/ \\overline{t\\_c (\\rho \\_i )} = 1/\\overline{\\beta\\_c(\\rho \\_i)},$$ \nwhere $\\overline{t \\_c (\\rho \\_i) } $ and $\\overline{\\beta\\_c(\\rho \\_i)}$ are the average of {*t**c*(*ρ**i*)}*i* = 1*N**c* and {*β**c*(*ρ**i*)}*i* = 1*N**c*, respectively. Thus, *β**c*(*ρ*) can be considered as an appropriate measure of local distinguishability in a original operational sense, and also as a function whose average gives an upper bound for the locally distinguishable sets of states. Therefore, we investigate the difference of local distinguishability of *ρ* by one-way LOCC POVM, two-way LOCC POVM, and separable POVM in terms of *β**c*(*ρ*) in the following sections.\nLocal discrimination by separable POVM\n======================================\nIn this section, we investigate the minimum type 2 error probability $\\beta\\_{\\rm sep}(\\rho)= \\frac{t\\_{{\\rm sep}}(\\rho)}{D}$ in terms of separable POVMs, which are given by {*N**i* ⊗ *M**i*}*i* with the conditions ∑*i**N**i* ⊗ *M**i* = *I*, *N**i* ≥ 0, and *M**i* ≥ 0. The main purpose of this section is proving the following theorem:\n[thm2] The inequality\n \n$$\\label{ineq-2}\nt\\_{{\\rm sep}}(\\rho) \\ge \n\\max\\{(\\Tr \\sqrt{\\rho\\_A})^2,(\\Tr \\sqrt{\\rho\\_B})^2\\}$$ \nholds for a bipartite state *ρ* on H*A* ⊗ H*B*, where *ρ**A* and *ρ**B* are the reduced density matrix of H*A* and H*B*, respectively. Any pure state satisfies its equality. In other words, the following inequality concerning the minimum error probability $\\beta \\_{\\rm sep}(\\rho)$ holds:\n \n$$\\beta\\_{{\\rm sep}}(\\rho) \\ge \n\\frac{1}{D}\\max\\{(\\Tr \\sqrt{\\rho\\_A})^2,(\\Tr \\sqrt{\\rho\\_B})^2\\}.$$ \nFor a bipartite pure state, the right-hand side of Eq.([ineq-2]) is proportional to an entanglement monotone called the global robustness of entanglement $R\\_g({{\\left \\vert}\\Psi {\\right \\rangle}})$ except an unimportant constant term.\nApplying Theorem [thm2] for Eq.([local dicrimination bound]), we can immediately derive the following corollary concerning the perfect discrimination of a given set of states in term of separable operations:\nIf a set of states {*ρ**i*}*i* = 1*N* is perfectly distinguishable by separable operations, then, the set of states {*ρ**i*}*i* = 1*N* satisfies the following inequality:\n \n$$\\label{ineq-owari}\nN \\le D/\\overline{\\max \\{ (\\Tr \\sqrt{\\rho \\_{iA}})^2, (\\Tr \\sqrt{\\rho\\_{iB}})^2 \\}},$$ \nwhere $\\overline{\\max \\{ (\\Tr \\sqrt{\\rho \\_{iA}})^2, (\\Tr \\sqrt{\\rho\\_{iB}})^2 \\}}$ is the average of $\\max \\{ (\\Tr \\sqrt{\\rho \\_{iA}})^2, (\\Tr \\sqrt{\\rho\\_{iB}})^2 \\}$ for all 1 ≤ *i* ≤ *N*.\nThe above inequality is weaker than the inequality ([local dicrimination bound]). However, the inequality ([ineq-owari]) is superior to the inequality ([local dicrimination bound]) in terms of the efficiency of the computation; that is, in general, we can not efficiently compute the bound in Eq.([local dicrimination bound]), since the function $t\\_{\\rm sep}(\\rho)$ includes the big variational problem.\nPure states case\n----------------\nFirst, for a technical reason, we concentrate the pure states case, and define a set of POVM elements $\\mathcal{T}\\_{\\widetilde{\\rm sep}}$ by,\n \n$$\\begin{aligned}\n\\mathcal{T} \\_{\\widetilde{\\rm sep}} \\stackrel{\\rm def}{=} \\left \\{ T \\left|\n\\nonumber \\quad T \\le I\\_{AB}, T = \\sum \\_i N \\_i \\otimes M\\_i,\\quad\n \\forall i, N\\_i \\ge 0, M\\_i \\ge 0 \\right.\\right \\}.\\end{aligned}$$ \n$\\mathcal{T} \\_{\\widetilde{\\rm sep}}$ is a set of POVM elements can be decomposed into a separable form; we say a positive linear operator *M* has a separable form, if $M/\\Tr M$ is a separable state. Since the definition of $\\mathcal{T}\n\\_{\\widetilde{\\rm sep}}$ is equivalent to the definition of $\\mathcal{T}\\_{\\rm sep}$ except the condition $I -T \\in \\mathcal{T}\\_{\\rm sep}$, $\\mathcal{T}\\_{\\rm sep}$ is a subset of $\\mathcal{T}\\_{\\widetilde{\\rm sep}} $. Note that even if $T \\in \\mathcal{T}\\_{\\widetilde{\\rm sep}}$, *I* − *T* does not necessary satisfy $I-T \\in \\mathcal{T}\\_{\\widetilde{\\rm sep}}$; that is, $\\mathcal{T}\\_{\\rm sep}$ does not coincide with $\\mathcal{T}\\_{\\widetilde{\\rm sep}} $. For example, suppose a set of states $\\{ {{\\left \\vert}\\Psi \\_i {\\right \\rangle}} \\}\\_{i=1}^{m} \\subset {\\mathcal{H}}\\_A \\otimes {\\mathcal{H}}\\_B$ (*m* < dimH) is an unextendable product basis, and a POVM *T* is defined as $T \\stackrel{\\rm def}{=} \\sum \\_i {{\\left \\vert}\\Psi \\_i {\\right \\rangle}}{{\\left \\langle}\\Psi \\_i {\\right \\vert}}$. Then, *T* belongs to $\\mathcal{T}\\_{\\widetilde{\\rm sep}} $, but not to $\\mathcal{T}\\_{\\rm sep}$ since $I-T = I - \\sum \\_{i=1}^m {{\\left \\vert}\\Psi \\_i {\\right \\rangle}} {{\\left \\langle}\\Psi \\_i {\\right \\vert}} $ is proportion to a (bound) entangled state, and does not a separable form. Similarly, we can define $t \\_{\\widetilde{\\rm sep}} $ as,\n \n$$\\begin{aligned}\n\\label{definition of t widetilde}\nt \\_{\\widetilde{\\rm sep}} (\\rho ) \\stackrel{\\rm def}{=} \\min \\left \\{ \\Tr T |\n T \\in \\mathcal{T}\\_{\\widetilde{\\rm sep}}, \\Tr \\rho T = 1 \\right \\}.\\end{aligned}$$ \nBy definition, $t \\_{\\widetilde{\\rm sep}} (\\rho )$ apparently gives a lower bound of $t\\_{\\rm sep} (\\rho) = d^2\\beta \\_{\\rm sep}(\\rho)$, that is, for all *ρ* ∈ *S*(H),\n \n$$\\label{inequality for t\\_sep }\nt\\_{\\widetilde{\\rm sep}} (\\rho) \\le t\\_{\\rm sep}(\\rho) \n= d^2 \\beta\\_{\\rm sep}(\\rho).$$ \nThen, we can see that $t \\_{\\widetilde{\\rm sep}} (\\rho) $ is actually equal to *d*(*ρ*) which is defined in Theorem 1 of the paper as:\n \n$$\\label{definition of d}\nd(\\rho) \\stackrel{\\rm def}{=} \\min \\left \\{ \\frac{1}{\\Tr \\rho \\omega} \\Big| 0 \\le \\frac{\\omega}{\\Tr \\rho \\omega} \\le I,\n\\omega \\in {\\rm SEP} \\right \\},$$ \nwhere ${\\rm SEP}$ is the set of all separable states. We can easily check this fact just by defining $T \\stackrel{\\rm def}{=} \\frac{\\omega}{\\Tr \\rho \\omega} $; then, *T* satisfies 0 ≤ *T* ≤ *I*, $\\Tr \\rho T = 1$, and $T \\in \\mathcal{T}\\_{\\widetilde{\\rm sep}}$. From Theorem 2 of the paper, for an arbitrary multipartite pure state ${{\\left \\vert}\\Psi {\\right \\rangle}}$, $t \\_{\\widetilde{\\rm sep}}(\\rho)$ satisfies the following inequality:\n \n$$\\label{separable formula}\nt\\_{\\widetilde{\\rm sep}} ({{\\left \\vert}\\Psi {\\right \\rangle}} ) = d({{\\left \\vert}\\Psi {\\right \\rangle}}) \\ge 1+\nR\\_g({{\\left \\vert}\\Psi {\\right \\rangle}}),$$ \nand $R\\_g({{\\left \\vert}\\Psi {\\right \\rangle}})$ is the global robustness of entanglement defined as:\n \n$$\\label{definition of robustness}\nR\\_g(\\rho) \\stackrel{\\rm def}{=} \\min \\left \\{ t \\ge 0 \\Big| \n\\exists {\\rm a} \\ {\\rm state}\\ \\Delta, \\ {\\rm s.t.}\\ \\frac{1}{1+t}(\\rho +t \\Delta ) \\in {\\rm SEP} \\right \\}.$$ \nFor a bipartite pure state, we can know a more detail of $t\\_{\\widetilde{\\rm sep}} ({{\\left \\vert}\\Psi {\\right \\rangle}} )$ as follows. First, it was proven that $t\\_{\\widetilde{\\rm sep}} ({{\\left \\vert}\\Psi {\\right \\rangle}} )$ coincides with the robustness of entanglement $R\\_g({{\\left \\vert}\\Psi {\\right \\rangle}}) $ for an arbitrary pure bipartite state ${{\\left \\vert}\\Psi {\\right \\rangle}}$. This fact can be seen by checking that the optimal states of *R**g*(*ρ*), which was derived in satisfies the condition of *d*(*ρ*); the optimal state of *R**g*(*ρ*) is also an optimal state of *d*(*ρ*). Moreover, we know that the value of $R\\_g({{\\left \\vert}\\Psi {\\right \\rangle}})$ is given by the following formula for a bipartite state ${{\\left \\vert}\\Psi {\\right \\rangle}}$ :\n \n$$R\\_g(\\rho) = \\Bigl(\\sum \\_{i} \\sqrt{\\lambda \\_i}\\Bigr)^2 -1,$$ \nwhere {*λ**i*}*i* = 1*d* is the Schmidt coefficients of ${{\\left \\vert}\\Psi {\\right \\rangle}}$; ${{\\left \\vert}\\Psi {\\right \\rangle}}$ can be decomposed as ${{\\left \\vert}\\Psi {\\right \\rangle}} = \\sum \\_i \\sqrt{\\lambda \\_i}\n{{\\left \\vert}e\\_i {\\right \\rangle}}\\otimes {{\\left \\vert}f\\_i {\\right \\rangle}}$ by choosing an appropriate orthonormal basis sets of the local Hilbert spaces $\\{ {{\\left \\vert}e\\_i {\\right \\rangle}} \\}\\_{i=1}^d \\subset {\\mathcal{H}}\\_A$ and $\\{ {{\\left \\vert}f\\_i {\\right \\rangle}} \\}\\_{i=1}^d \\subset {\\mathcal{H}}\\_B$. Thus, we derive\n[lemma for bipartite twidetilde] For a bipartite pure state ${{\\left \\vert}\\Psi {\\right \\rangle}} \\in {\\mathcal{H}}\\_A \\otimes {\\mathcal{H}}\\_B$,\n \n$$\\label{bipartite t\\_widetilde}\nt\\_{\\widetilde{\\rm sep}} ({{\\left \\vert}\\Psi {\\right \\rangle}} ) = d({{\\left \\vert}\\Psi {\\right \\rangle}}) = 1+\nR\\_g({{\\left \\vert}\\Psi {\\right \\rangle}}) = \\Bigl(\\sum \\_{i} \\sqrt{\\lambda \\_i}\\Bigr)^2.$$ \nAlthough this lemma is a known result, as a preparation of the proof of the next theorem, we give a complete proof of Eq.([bipartite twidetilde]), in which we prove directly the equation $t\\_{\\widetilde{\\rm sep}} ({{\\left \\vert}\\Psi {\\right \\rangle}} ) = \\Bigl(\\sum \\_{i} \\sqrt{\\lambda \\_i}\\Bigr)^2$ from the definition of $t\\_{\\widetilde{\\rm sep}} ({{\\left \\vert}\\Psi {\\right \\rangle}} )$.\nThis proof is divided into two-steps. In the first step, we prove that $\\Bigl(\\sum \\_{i} \\sqrt{\\lambda \\_i}\\Bigr)^2$ is the lower bound of $t\\_{\\widetilde{\\rm sep}}({{\\left \\vert}\\Psi {\\right \\rangle}})$. Then, in the second step, we construct POVM element *T* which attains this lower bound. For convenience, we define ${{\\left \\vert}M \\_{\\Psi} {\\right \\rangle}} = \\frac{1}{\\sqrt{d}}\\sum \\_{i=1}^d\n{{\\left \\vert}e\\_i {\\right \\rangle}} \\otimes {{\\left \\vert}f\\_i {\\right \\rangle}}$ where $\\{ {{\\left \\vert}e\\_i {\\right \\rangle}} \\}\\_{i=1}^d$ and $\\{\n{{\\left \\vert}f\\_i {\\right \\rangle}} \\}\\_{i=1}^d$ are the Schmidt basis of ${{\\left \\vert}\\Psi {\\right \\rangle}}$; thus, ${{\\left \\vert}M\n\\_{\\Psi} {\\right \\rangle}}$ is the maximum entangled state sharing the Schmidt basis with ${{\\left \\vert}\\Psi {\\right \\rangle}}$. Then, we derive $d | {{\\left \\langle}M\\_{\\Psi}\n \\vert \\Psi {\\right \\rangle}} |^2 = \\Bigl(\\sum \\_{i} \\sqrt{\\lambda \\_i}\\Bigr)^2$.\nAs the first step, we prove the following inequality;\n \n$$\\begin{aligned}\nt\\_{\\widetilde{\\rm sep}} ({{\\left \\vert}\\Psi {\\right \\rangle}}) &=& \\min \\{ \\Tr T | 0 \\le T \\le I,\\ T \\ {\\rm is \\ sep}, \\ {{\\left \\langle}\\Psi {\\right \\vert}}T {{\\left \\vert}\\Psi {\\right \\rangle}} =1 \\} \\nonumber \\\\\n &\\ge & \\min \\{ d {{\\left \\langle}M\\_{\\Psi} {\\right \\vert}} T {{\\left \\vert}M\\_{\\Psi} {\\right \\rangle}} | 0 \\le T \\le I, T \\ {\\rm is \\ sep}, {{\\left \\langle}\\Psi {\\right \\vert}}T{{\\left \\vert}\\Psi {\\right \\rangle}} =1 \\} \\label{1stineq} \\nonumber \\\\\n &\\ge & d | {{\\left \\langle}M\\_{\\Psi} \\vert \\Psi {\\right \\rangle}} |^2 \\label{2ndineq}.\\end{aligned}$$ \nTo prove the first inequality ([1stineq]), since both $\\Tr T$ and $d {{\\left \\langle}M\\_{\\Psi} {\\right \\vert}} T {{\\left \\vert}M\\_{\\Psi} {\\right \\rangle}}$ are linear for *T*, it is enough to prove only in the case that *T* can be written down as $T = {{\\left \\vert}a {\\right \\rangle}}{{\\left \\langle}a {\\right \\vert}} \\otimes {{\\left \\vert}b {\\right \\rangle}}{{\\left \\langle}b {\\right \\vert}}$ by using un-normalized vectors ${{\\left \\vert}a {\\right \\rangle}}$ and ${{\\left \\vert}b {\\right \\rangle}}$. Suppose ${{\\left \\vert}a {\\right \\rangle}} = \\sum \\_{i=1}^d \\alpha \\_i {{\\left \\vert}e\\_i {\\right \\rangle}}$ and ${{\\left \\vert}b {\\right \\rangle}} = \\sum \\_{i=1}^d \\beta \\_i {{\\left \\vert}f\\_i {\\right \\rangle}}$. Then, using Schwarz’s inequality, we can prove as follows\n \n$$\\begin{aligned}\n\\Tr T = ( \\sum \\_i | \\alpha \\_i |^2)( \\sum \\_i | \\beta \\_i |^2) \n \\ge |\\sum \\_j \\alpha \\_j \\beta \\_j |^2 \n = d {{\\left \\langle}M\\_{\\Psi} {\\right \\vert}} T {{\\left \\vert}M\\_{\\Psi} {\\right \\rangle}}.\\end{aligned}$$ \nFor the second inequality ([2ndineq]), since the relations ${{\\left \\langle}\\Psi {\\right \\vert}}T{{\\left \\vert}\\Psi {\\right \\rangle}} =1$ and *T* ≤ *I* deduce that ${{\\left \\vert}\\Psi {\\right \\rangle}}$ is an eigenvector of the largest eigenvalue 1 of *T*, we derive $T \\ge {{\\left \\vert}\\Psi {\\right \\rangle}}\n{{\\left \\langle}\\Psi {\\right \\vert}}$. Therefore, the inequality $d {{\\left \\langle}M\\_{\\Psi} {\\right \\vert}} T {{\\left \\vert}M\\_{\\Psi} {\\right \\rangle}} \\ge\nd|{{\\left \\langle}M\\_{\\Psi} \\vert \\Psi {\\right \\rangle}} |^2$ is derived by taking the mean value with respect to ${{\\left \\vert}M\n\\_{\\Psi} {\\right \\rangle}}$.\nAs the second step, we construct an example of POVM element *T* which achieves the lower bound we derived above. Define *T*0 as $T\\_0 = {{\\left \\vert}a {\\right \\rangle}} {{\\left \\langle}a {\\right \\vert}} \\otimes {{\\left \\vert}b {\\right \\rangle}}{{\\left \\langle}b {\\right \\vert}}$ where ${{\\left \\vert}a {\\right \\rangle}} =\n\\sum \\_{i=1}^d (\\lambda \\_i)^{1/4} {{\\left \\vert}e\\_i {\\right \\rangle}}$ and ${{\\left \\vert}b {\\right \\rangle}} = \\sum \\_{i=1}^d\n(\\lambda \\_i)^{1/4}{{\\left \\vert}f\\_i {\\right \\rangle}}$; then, *T*0 satisfies $\\Tr T\\_0 = \n\\Bigl(\\sum \\_{i} \\sqrt{\\lambda \\_i}\\Bigr)^2$. Moreover, since $P\\_0 T\\_0 P\\_0 =\n{{\\left \\vert}\\Psi {\\right \\rangle}} {{\\left \\langle}\\Psi {\\right \\vert}}$ where $P\\_0 = \\sum \\_{i=1}^d {{\\left \\vert}e\\_i {\\right \\rangle}}\n{{\\left \\langle}e\\_i {\\right \\vert}} \\otimes {{\\left \\vert}f\\_i {\\right \\rangle}} {{\\left \\langle}f\\_i {\\right \\vert}}$, *T*0 satisfies ${{\\left \\langle}\\Psi {\\right \\vert}} T\\_0 {{\\left \\vert}\\Psi {\\right \\rangle}} = 1$. Since *T*0 apparently satisfies 0 ≤ *T*0, the inequality *T* ≤ *I* is the only remaining condition which the optimal POVM element *T* attaining the equality $\\Tr T = t\\_{\\widetilde{\\rm sep}} ({{\\left \\vert}\\Psi {\\right \\rangle}})$ must satisfy. Since *T*0 does not generally satisfies the inequality *T*0 ≤ *I*, we construct a new POVM element *T* which satisfies 0 ≤ *T* ≤ *I* from *T*0. In order to construct the POVM element *T* from *T*0, we use twirling technique here. We define a family of local unitary operators *U**θ⃗* parameterized by *θ⃗* = {*θ**i*}*i* = 1*d* as follows,\n \n$$\\label{definition of U\\_theta}\nU\\_{\\overrightarrow{\\theta}} = (\\sum\\_{j=1}^d e^{i \\theta \\_j} {{\\left \\vert}e\\_j {\\right \\rangle}}{{\\left \\langle}e\\_j {\\right \\vert}})\n\\otimes (\\sum \\_{k=1}^d e^{-i \\theta \\_k} {{\\left \\vert}f\\_k {\\right \\rangle}}{{\\left \\langle}f\\_k {\\right \\vert}} ).$$ \nNote that (H⊗ 2, *U**θ⃗*) is a unitary representation of the compact topological group $\\overbrace{U(1) \\times \\cdots \\times U(1)}^n $; by means of a unitary representation of a compact topological group, we implement the “*twirling*” operation (the averaging over the compact topological group) for a state (or POVM). Then, we define *T* as the operator which is constructed by twirling *U**θ⃗**T*0*U**θ⃗*† over parameters *θ⃗* = {*θ**i*}*i* = 1*d*. Since by an action of twirling operation, a given state is projected to the subspace of all invariant elements of the group action, we can calculate *T* as follows:\n \n$$\\begin{aligned}\nT & \\stackrel{\\rm{ def} }{=} & \\int \\_0^{2 \\pi} \\cdots \\int \\_0^{2\n\\pi} U\\_{\\overrightarrow{\\theta}} T\\_0 U\\_{\\overrightarrow{\\theta}}^{\\dagger} d \\theta \\_1 \\cdots d \\theta \\_d \\\\\n&=& (\\sum \\_{j=1}^d {{\\left \\vert}e\\_j {\\right \\rangle}}{{\\left \\langle}e\\_j {\\right \\vert}} \\otimes {{\\left \\vert}f\\_j {\\right \\rangle}}{{\\left \\langle}f\\_j {\\right \\vert}} )T\\_0 ( \\sum \\_{j=1}^d {{\\left \\vert}e\\_j {\\right \\rangle}}{{\\left \\vert}e\\_j {\\right \\rangle}} \\otimes {{\\left \\vert}f\\_j {\\right \\rangle}}{{\\left \\langle}f\\_j {\\right \\vert}}) \\\\\n&\\ & \\quad +\\sum \\_{j \\neq k} \\left ( {{\\left \\vert}e\\_j {\\right \\rangle}}{{\\left \\langle}e\\_j {\\right \\vert}}\\otimes {{\\left \\vert}f\\_k {\\right \\rangle}}{{\\left \\langle}f\\_k {\\right \\vert}} \\right ) T\\_0 \\left ( {{\\left \\vert}e\\_j {\\right \\rangle}}{{\\left \\langle}e\\_j {\\right \\vert}}\\otimes {{\\left \\vert}f\\_k {\\right \\rangle}}{{\\left \\langle}f\\_k {\\right \\vert}} \\right ) \\\\ \n& = & (\\sum \\_{i=1}^d \\sqrt{\\lambda \\_i} {{\\left \\vert}e\\_i {\\right \\rangle}} \\otimes {{\\left \\vert}f\\_i {\\right \\rangle}})\n (\\sum \\_{i=1}^d \\sqrt{\\lambda \\_i} {{\\left \\langle}e\\_i {\\right \\vert}} \\otimes {{\\left \\langle}f\\_i {\\right \\vert}} ) \\nonumber \\\\\n & \\ & \\quad + \\sum \\_{i \\neq j} \\sqrt{\\lambda \\_i \\lambda \\_j} {{\\left \\vert}e\\_i {\\right \\rangle}}{{\\left \\langle}e\\_i {\\right \\vert}}\n \\otimes {{\\left \\vert}f\\_j {\\right \\rangle}}{{\\left \\langle}f\\_j {\\right \\vert}}.\\end{aligned}$$ \nSince $\\sqrt{\\lambda \\_i \\lambda \\_j} \\le 1$, *T* ≤ *I*. Moreover, *T* satisfies 0 ≤ *T* ≤ *I*, ${{\\left \\langle}\\Psi {\\right \\vert}} T {{\\left \\vert}\\Psi {\\right \\rangle}} =1$, and is in the separable form; we only applied the local unitary *U**θ⃗* to un-normalized product state *T*0, and, then, took an average over parameters *θ⃗*. Thus, we derive the inequality $t\\_{\\widetilde{\\rm sep}}({{\\left \\vert}\\Psi {\\right \\rangle}}) \\le \\Tr T = \\Bigl(\\sum \\_{i} \\sqrt{\\lambda \\_i}\\Bigr)^2$. Since we have already proven the converse inequality, we conclude $t\\_{\\widetilde{\\rm sep}}({{\\left \\vert}\\Psi {\\right \\rangle}}) = \\Bigl(\\sum \\_{i} \\sqrt{\\lambda \\_i}\\Bigr)^2$ \n▫\nFinally, by means of Lemma [lemma for bipartite twidetilde], we can derive the following theorem, i.e., Theorem [thm2] in the pure states case:\n[theorem separable main] For a bipartite pure state ${{\\left \\vert}\\Psi {\\right \\rangle}}$,\n \n$$\\begin{aligned}\n\\label{main theorem of sep POVM}\n\\fl \\beta\\_{\\rm sep} ({{\\left \\vert}\\Psi {\\right \\rangle}} )=\\frac{1}{d^2}t\\_{\\rm sep}({{\\left \\vert}\\Psi {\\right \\rangle}}) &=& \\frac{1}{d^2}\\left (1+\nR\\_g({{\\left \\vert}\\Psi {\\right \\rangle}}) \\right) \\nonumber \\\\\n&=& \\frac{1}{d^2}\\Bigl(\\sum \\_{i} \\sqrt{\\lambda \\_i}\\Bigr)^2 = \\frac{1}{d^2}(\\Tr \\sqrt{\\rho \\_A})^2 = \\frac{1}{d^2}(\\Tr \\sqrt{\\rho \\_B})^2,\\end{aligned}$$ \nwhere {*λ**i*}*i* = 1*d* is the Schmidt coefficients of ${{\\left \\vert}\\Psi {\\right \\rangle}}$.\nSince by the definition $t\\_{\\widetilde{\\rm sep}} ({{\\left \\vert}\\Psi {\\right \\rangle}} ) \\le t\\_{\\rm sep} ({{\\left \\vert}\\Psi {\\right \\rangle}} ) $, all what we need to prove is that the optimal POVM *T* for $t\\_{\\widetilde{\\rm sep}} ({{\\left \\vert}\\Psi {\\right \\rangle}} )$ is also the optimal POVM for $t\\_{\\rm sep} ({{\\left \\vert}\\Psi {\\right \\rangle}} )$; that is, *I* − *T* also has a separable form.\nAs we have already shown in the proof of Lemma [lemma for bipartite twidetilde], the optimal POVM *T* for $t\\_{\\widetilde{\\rm sep}} ({{\\left \\vert}\\Psi {\\right \\rangle}} )$ can be written down as\n \n$$\\label{optimal POVM element separable}\n\\fl T = (\\sum \\_{i=1}^d \\sqrt{\\lambda \\_i} {{\\left \\vert}e\\_i {\\right \\rangle}} \\otimes {{\\left \\vert}f\\_i {\\right \\rangle}})\n (\\sum \\_{i=1}^d \\sqrt{\\lambda \\_i} {{\\left \\langle}e\\_i {\\right \\vert}} \\otimes {{\\left \\langle}f\\_i {\\right \\vert}} ) \n~ + \\sum \\_{i \\neq j} \\sqrt{\\lambda \\_i \\lambda \\_j} {{\\left \\vert}e\\_i {\\right \\rangle}}{{\\left \\langle}e\\_i {\\right \\vert}}\n \\otimes {{\\left \\vert}f\\_j {\\right \\rangle}}{{\\left \\langle}f\\_j {\\right \\vert}},$$ \nwhere $\\{ {{\\left \\vert}e\\_i {\\right \\rangle}}\\otimes {{\\left \\vert}f\\_j {\\right \\rangle}} \\}\\_{ij}$ and {*λ**i*}*i* = 1*d* are the Schmidt basis and the Schmidt coefficients corresponding to ${{\\left \\vert}\\Psi {\\right \\rangle}} = \\sum \\_{i=1}^d \\sqrt{\\lambda \\_i} {{\\left \\vert}e\\_if\\_j {\\right \\rangle}}$, respectively. Suppose\n \n$$\\fl \\overline{T\\_0} \\stackrel{\\rm def}{=} \\frac{1}{2}\\sum \\_{i\\neq j}{{\\left \\vert}\\overline{a\\_{ij}} {\\right \\rangle}}{{\\left \\langle}\\overline{a\\_{ij}} {\\right \\vert}} \n\\otimes {{\\left \\vert}\\overline{b\\_{ij}} {\\right \\rangle}}{{\\left \\langle}\\overline{b\\_{ij}} {\\right \\vert}}\n+ \\sum \\_{i\\neq j}\\{ \\sum \\_{k \\neq i,j} \\lambda \\_k + (\\sqrt{\\lambda \\_i} - \\sqrt{\\lambda \\_j})^2\\}{{\\left \\vert}e\\_if\\_j {\\right \\rangle}} {{\\left \\langle}e\\_if\\_j {\\right \\vert}},$$ \nwhere ${{\\left \\vert}\\overline{a\\_{ij}} {\\right \\rangle}}$ and ${{\\left \\vert}\\overline{b\\_{ij}} {\\right \\rangle}}$ are defined as ${{\\left \\vert}\\overline{a\\_{ij}} {\\right \\rangle}} \\stackrel{\\rm def}{=} (\\lambda \\_j)^{\\frac{1}{4}}{{\\left \\vert}e\\_i {\\right \\rangle}}-(\\lambda \\_i)^{\\frac{1}{4}}{{\\left \\vert}e\\_j {\\right \\rangle}}$ and ${{\\left \\vert}\\overline{b\\_{ij}} {\\right \\rangle}} \\stackrel{\\rm def}{=} (\\lambda \\_j)^{\\frac{1}{4}}{{\\left \\vert}f\\_i {\\right \\rangle}}+(\\lambda \\_i)^{\\frac{1}{4}}{{\\left \\vert}f\\_j {\\right \\rangle}}$ for *i* ≠ *j*, respectively. Then, as is proven in [appendix], the relation\n \n$$\\int \\_0^{2 \\pi} \\cdots \\int \\_0^{2\n\\pi} U\\_{\\overrightarrow{\\theta}} \\overline{T\\_0} U\\_{\\overrightarrow{\\theta}}^{\\dagger} d \\theta \\_1 \\cdots d \\theta \\_d\n=I-T\n\\label{average}$$ \nholds, where a local unitary operator *U**θ⃗* is defined as Eq.([definition of Utheta]). By the definition, $\\int \\_0^{2 \\pi} \\cdots \\int \\_0^{2\n\\pi} U\\_{\\overrightarrow{\\theta}} \\overline{T\\_0} U\\_{\\overrightarrow{\\theta}}^{\\dagger} d \\theta \\_1 \\cdots d \\theta \\_d$ is apparently a separable POVM element. Therefore, we can conclude the equality $t\\_{\\widetilde{\\rm sep}} ({{\\left \\vert}\\Psi {\\right \\rangle}} ) = t\\_{\\rm sep} ({{\\left \\vert}\\Psi {\\right \\rangle}} ) $ for a bipartite pure state. By means of Lemma [lemma for bipartite twidetilde], we derive Eq.([main theorem of sep POVM]). ▫\nThus, $t\\_{\\rm sep} ({{\\left \\vert}\\Psi {\\right \\rangle}} )$ is equivalent to $1+ R\\_g({{\\left \\vert}\\Psi {\\right \\rangle}})$, and the minimum type 2 error probability $\\beta\\_{\\rm sep}({{\\left \\vert}\\Psi {\\right \\rangle}}) $ only depends on the global robustness of entanglement $R\\_g({{\\left \\vert}\\Psi {\\right \\rangle}})$ for a bipartite pure state ${{\\left \\vert}\\Psi {\\right \\rangle}}$. In this case, the optimal POVM {*T*, *I* − *T*} can be derived by using Eq.([optimal POVM element separable]) as the definition of the POVM element *T*.\nWe should note that Theorem [theorem separable main] not only gives a way to calculate the minimum type 2 error probability under separable operations $\\beta\\_{\\rm sep}({{\\left \\vert}\\Psi {\\right \\rangle}})$, but this theorem gives a complete relationship between the local distinguishability of a bipartite state under separable operations and the entanglement of the state. In the previous paper, it was shown the global robustness of entanglement $R\\_g({{\\left \\vert}\\Psi {\\right \\rangle}})$ gives an upper-bound for the maximum number of distinguishable states under separable operations. However, the present result shows that $R\\_g({{\\left \\vert}\\Psi {\\right \\rangle}})$ is nothing but the local distinguishability (against the completely mixed state) itself at least for a bipartite pure state. In other words, it is shown that robustness of entanglement has rigorously operational meaning for bipartite pure states in terms of the local discrimination from the completely mixed state $\\rho\\_{\\rm mix}$.\nMixed states case\n-----------------\nNow, we prove Theorem [thm2] for a general mixed bipartite state.\n[Theorem [thm2]] First we prove the inequality $t\\_{\\rm sep}(\\rho)\\ge (\\Tr \\sqrt{\\rho\\_A})^2$. Adding the system *B*ʹ, we choose a purification ∣Φ⟩ of *ρ*. In the following, we will prove the inequality $t\\_{\\widetilde{\\rm sep}}(\\rho) \\ge t\\_{\\widetilde{\\rm sep}}({{\\left \\vert}\\Phi {\\right \\rangle}})$. If this inequality holds, applying Eq.([inequality for tsep ]) and Eq.([bipartite twidetilde]), we obtain $t\\_{\\rm sep}(\\rho)\\ge t\\_{\\widetilde{\\rm sep}}(\\rho) \\ge t\\_{\\widetilde{\\rm sep}}({{\\left \\vert}\\Phi {\\right \\rangle}}) = (\\Tr \\sqrt{\\rho\\_A})^2$.\nDefine a separable positive operator *T* = ∑*i**p**i*∣*e**i*⟩⟨*e**i*∣ ⊗ ∣*f**i*⟩⟨*f**i*∣ (*e**i* ∈ H*A*, *f**i* ∈ H*B*, ∥*f**i*∥ = 1, ∥*e**i*∥ = 1) such that 0 ≤ *T* ≤ *I**A**B* and $\\Tr \\rho T=1$. Thus, $\\langle \\Phi| \\sum\\_{i}p\\_i \n\\bigl(|e\\_i\\rangle\\langle e\\_i|\\otimes |f\\_i\\rangle\\langle f\\_i|\\otimes I\\_{B'} \\bigr)\n|\\Phi\\rangle=1$. Now, we focus on the bipartite system *A* and *B**B*ʹ. Then, we choose the state ∣*f̃**i*⟩ (∥*f̃**i*∥ = 1) on H*B**B*ʹ such that $\\Tr\\_{A} (|\\Phi\\rangle\\langle \\Phi |) \n(|e\\_i\\rangle\\langle e\\_i|\\otimes I\\_{BB'})=\nc\\_i |\\tilde{f}\\_i\\rangle\\langle \\tilde{f}\\_i|$, where *c**i* is the normalizing constant. Define the state ∣*f**i*ʹ⟩ (∥*f**i*ʹ∥ = 1) on H*B**B*ʹ by\n \n$$|f\\_i'\\rangle \\langle f\\_i'|\n:= \n\\frac{1}{\\langle \\tilde{f}\\_i|P\\_i|\\tilde{f}\\_i\\rangle }\nP\\_i |\\tilde{f}\\_i\\rangle\\langle \\tilde{f}\\_i|P\\_i \n\\le P\\_i,$$ \nwhere the projection *P**i* is defined by *P**i* : = ∣*f**i*⟩⟨*f**i*∣ ⊗ *I**B*ʹ. Since ⟨*f̃**i*∣*P**i*∣*f̃**i*⟩ = ⟨*f̃**i*∣*f**i*ʹ⟩⟨*f**i*ʹ∣*f̃**i*⟩,\n \n$$\\begin{aligned}\n\\fl \\Tr \\rho (|e\\_i\\rangle\\langle e\\_i|\\otimes |f\\_i\\rangle\\langle f\\_i|)\n=\\langle \\Phi|( |e\\_i\\rangle\\langle e\\_i|\n\\otimes |f\\_i\\rangle\\langle f\\_i|\\otimes I\\_{B'}|)|\\Phi\\rangle\n=\\langle \\Phi| (|e\\_i\\rangle\\langle e\\_i|\n\\otimes |f\\_i'\\rangle\\langle f\\_i'|)|\\Phi\\rangle.\\end{aligned}$$ \nThus, the relations\n \n$$\\begin{aligned}\nT':= \\sum\\_{i} p\\_i |e\\_i\\rangle\\langle e\\_i|\n\\otimes |f\\_i'\\rangle\\langle f\\_i'|\n\\le \n\\sum\\_{i} p\\_i |e\\_i\\rangle\\langle e\\_i|\n\\otimes |f\\_i\\rangle\\langle f\\_i| \\otimes I\\_{B'}\n\\le I\\_{ABB'}\\\\\n\\langle \\Phi |T'|\\Phi\\rangle=\\Tr \\rho T=1\\end{aligned}$$ \nhold. Moreover, *T*ʹ satisfies the equality $\\Tr T' = \\sum\\_{i} p\\_i = \\Tr T$. Thus, the inequality $t\\_{\\widetilde{\\rm sep}}(\\rho) \\ge t\\_{\\widetilde{\\rm sep}}({{\\left \\vert}\\Phi {\\right \\rangle}})$ holds. Therefore, the relations $t\\_{\\rm sep}(\\rho)\\ge t\\_{\\widetilde{\\rm sep}}(\\rho) \\ge t\\_{\\widetilde{\\rm sep}}({{\\left \\vert}\\Phi {\\right \\rangle}}) = (\\Tr \\sqrt{\\rho\\_A})^2$ hold.\nSimilarly, we can show the inequality $t\\_{\\rm sep}(\\rho)\\ge (\\Tr \\sqrt{\\rho\\_B})^2$. Thus, we obtain ([ineq-2]) in the mixed states case. ▫\nLocal discrimination by one-way LOCC\n====================================\nIn this section, we prove the following theorem concerning the local discrimination problem in terms of one-way LOCC in the direction *A* → *B*:\n[theorem one-way main] The inequality\n \n$$\\label{ineq-1}\nt\\_{\\rightarrow}(\\rho)\n\\ge {\\mathop{\\rm rank}}\\rho\\_A$$ \nholds for a bipartite state *ρ* on H*A* ⊗ H*B*. Any maximally correlated state *ρ* satisfies the equality. In other words, the following inequality concerning the minimum error probability *β*→(*ρ*) holds:\n \n$$\\beta\\_{\\rightarrow}(\\rho)\n\\ge \\frac{1}{D}{\\mathop{\\rm rank}}\\rho\\_A$$ \nIn the above theorem, a maximally correlated state is defined as a state which can be decompose in the following form:\n \n$$\\label{def of maximally correlated} \n\\rho = \\sum \\_{1 \\le i,j \\le d} \\alpha \\_{ij}{{\\left \\vert}u\\_i, v\\_i {\\right \\rangle}}{{\\left \\langle}u\\_j, v\\_j {\\right \\vert}},$$ \nwhere $\\{ {{\\left \\vert}u\\_i {\\right \\rangle}} \\}\\_{i=1}^d$ and $\\{ {{\\left \\vert}v\\_j {\\right \\rangle}} \\}\\_{j=1}^d$ are orthonormal bases of H*A* and H*B*, respectively ; apparently, a pure state is a maximally correlated state. Thus, $t\\_{\\rightarrow}({{\\left \\vert}\\Psi {\\right \\rangle}})=D \\beta\\_{\\rightarrow }({{\\left \\vert}\\Psi {\\right \\rangle}} )$ is equal to the Schmidt rank of a state for a bipartite pure state ${{\\left \\vert}\\Psi {\\right \\rangle}}$. In the case when *ρ* is a maximally correlated state satisfying Eq.([def of maximally correlated]), the optimal way to discriminate between *ρ* and the completely mixed state is the following: Suppose there are two parties called Alice and Bob. Both Alice and Bob measure their local states H*A* and H*B* in the bases $\\{ {{\\left \\vert}u\\_i {\\right \\rangle}} \\}\\_{i=1}^d$ and $\\{ {{\\left \\vert}v \\_j {\\right \\rangle}} \\}\\_{j=1}^d$, respectively, (of course, they only need to detect the support of the local states). Then, Alice informs her measurement result to Bob. Suppose Alice’s result is ${{\\left \\vert}u\\_k {\\right \\rangle}}$ and Bob’s result is ${{\\left \\vert}v\\_l {\\right \\rangle}}$. If *k* is equal to *l*, then, they judge that the given state is *ρ*. Otherwise, they judge that the given state is the completely mixed state.\nBy comparing Theorem [thm2] and Theorem [theorem one-way main], we can easily see that if a bipartite pure state ${{\\left \\vert}\\Psi {\\right \\rangle}}$ is not a maximally entangled state nor a product state, then, the strict inequality $ \\beta\\_{\\rm sep}\n({{\\left \\vert}\\Psi {\\right \\rangle}}) < \\beta\\_{\\rightarrow } ({{\\left \\vert}\\Psi {\\right \\rangle}})$ holds. Thus, we can conclude that there is a gap between the one-way local distinguishability and the separable local distinguishability for a bipartite pure state at least in the present problem settings from these results.\nApplying Theorem [theorem one-way main] for Eq.([local dicrimination bound]), we can extend Cohen’s characterization concerning the perfect discrimination of a given set of pure states in term of one-way LOCC to a set of mixed states:\nIf a set of bipartite states {*ρ**i*}*i* = 1*N* on H*A* ⊗ H*B* is perfectly distinguishable by one-way LOCC, then,\n \n$$\\label{ineq-one-way}\n\\sum \\_{i=1}^N {\\rm rank} \\rho \\_{iA} \\le D.$$ \nThis bound of the size of locally distinguishable sets for one-way LOCC is much stronger than the known bound for separable operations.\nAs a preparation for our proof of Theorem [theorem one-way main], we see the fact that there are several equivalent representations of the definition of one-way LOCC POVM elements. We start from the following representation which we can see immediately from the definition; that is, in a bipartite system H = H*A* ⊗ H*B*, if *T* ∈ T→, there exist sets of positive operators {*M**i*}*i* and {*N**j**i*}*j* such that\n \n*T* = ∑*i**j**M**i* ⊗ *N**j**i*, \n∑*i**M**i* ≤ *I**B*, and ∑*j**N**j**i* ≤ *I**A*, where {*M**i*}*i* is the POVM of the local measurement on H*A* and {*N**j**i*}*j* is the POVM of the local measurement on H*B* depending on the first measurement result *i*. Further, redefining *N**j**i* as *N**i* = ∑*j**N**j**i*, we derive the following equivalence relation:\n \n$$\\begin{aligned}\n& \\quad & T \\in \\mathcal{T}\\_{\\rightarrow} \\nonumber \\\\\n& \\Longleftrightarrow & \\exists \\{ M\\_i \\} \\_i \\ \\mathrm{and}\n\\nonumber \\ \\{ N \\_i \\} \\_i \\\\ &\\quad & s.t. \\forall i, 0 \\le M \\_i,\n0 \\le N\\_i \\le I\\_B, \\sum \\_i M \\_i \\le I\\_A,\n \\mathrm{and } \\ T = \\sum \\_{i} M\\_i \\otimes N \\_i.\\label{first formulation}\\end{aligned}$$ \nUsing this characterization, we obtain the following lemma.\n[probability] A one-way LOCC POVM element *T* = ∑*i**j**M**i* ⊗ *N**j**i* ∈ T→ satisfies $\\Tr \\rho T =1$ if and only if $\\Tr ( \\rho \\_A \\sum M\\_i ) = 1$ and $\\Tr (\\rho \\_{B, M\\_i} \\sum\n\\_j N\\_j^i ) = 1$ for all *i*, where $\\rho \\_A \\stackrel{\\rm def}{=}\n\\Tr \\_B \\rho$ and $\\rho \\_{B, M\\_i} \\stackrel{\\rm def}{=} \\Tr \\_A \\rho\nM\\_i \\otimes I \\_B / \\Tr \\rho M\\_i \\otimes I \\_B$.\nWe can calculate $\\Tr \\rho T$ as follows:\n \n$$\\begin{aligned}\n\\Tr \\rho T = \\sum \\_{ij} \\Tr \\rho M\\_i \\otimes N\\_j^i\n&=& \\sum \\_{ij} \\Tr \\{ (\\Tr \\_A \\rho (M\\_i \\otimes I\\_B) ) N\\_j^i \\} \\\\\n&=& \\sum \\_i \\Tr \\rho \\_A M\\_i \\cdot \\Tr \\rho \\_{B, M\\_i } (\\sum \\_j\nN\\_i^j) =1.\\end{aligned}$$ \nSince $\\sum \\_i \\Tr \\rho \\_A M\\_i \\le 1$ and $\\Tr \\rho \\_{B, M\\_i}\n(\\sum \\_j N\\_j^i) \\le 1$ for all *i*, we derive $\\sum \\_i \\Tr \\rho \\_A\nM\\_i = 1$ and $\\Tr \\rho \\_{B, M\\_i} (\\sum \\_j N\\_j^i) = 1$. The opposite direction is trivial. ▫\nNow, we prove Theorem [theorem one-way main] using the above lemma.\n[Theorem [theorem one-way main]] In order to detect a state perfectly, we need to detect the reduced density operator of the local system *A*, *ρ**A* as well as that of the other local system *B*, *ρ**B*, *M**i*, perfectly in each step. Thus, we can assume that *N**i* is a projection on *B* without loss of generality. Hence, $\\Tr T = \\sum\\_i \\Tr M\\_i \\cdot \\Tr N\\_i \\ge \\sum\\_i \\Tr M\\_i$. Since we have to detect the reduced density operator of the local system *A*, *ρ**A*, we obtain $\\Tr \\sum\\_i M\\_i\\rho\\_A= 1$, i.e., ([ineq-1]). When the state *ρ* is a maximally mixed state ∑1 ≤ *i*, *j* ≤ *d**a**i*, *j*∣*u**i*, *v**i*⟩⟨*u**j*, *v**j*∣, the reduced density *ρ**A* is ∑*i* = 1*d**a**i*, *i*∣*u**i*⟩⟨*u**i*∣. Thus, ${\\mathop{\\rm rank}}\\rho\\_A= d$. In this case, we can perfectly detect this state by the one-way LOCC test ∑*i* = 1*d*∣*u**i*⟩⟨*u**i*∣ ⊗ ∣*v**i*⟩⟨*v**i*∣. ▫\nWe should note the following fact: Although a maximally correlated state satisfies the equality of Eq.([ineq-1]), the converse is not necessarily true. Even if *v**i* is not orthogonal, we can perfectly detect this state by the one-way LOCC test ∑*i* = 1*d*∣*u**i*⟩⟨*u**i*∣ ⊗ ∣*v**i*⟩⟨*v**i*∣. When the rank of the state ∑1 ≤ *i*, *j* ≤ *d**a**i*, *j*⟨*v**j*|*v**i*⟩∣*u**i*⟩⟨*u**j*∣ is *d*, the rank of *ρ**A* is *d*. That is, this gives a counter example of the converse.\nLocal discrimination by two-way LOCC\n====================================\nSo far, we have calculated the minimum error probability of the local discrimination problem for one-way LOCC $\\beta\\_{\\rightarrow}({{\\left \\vert}\\Psi {\\right \\rangle}})$, and separable operations $\\beta\\_{\\rm sep}({{\\left \\vert}\\Psi {\\right \\rangle}})$. In this section, we focus on discrimination protocols by two-way LOCC. Since the two-way LOCC is mathematically complicated, it is difficult to derive the minimum two-way LOCC discrimination protocol, and as a result, it is difficult to derive the exact value of $\\beta\\_{\\leftrightarrow}({{\\left \\vert}\\Psi {\\right \\rangle}})$. However, in order to show the difference of the efficiency of one-way and two-way local discrimination protocols, (which is actually our main purpose of this paper,) it is enough to find the upper-bound of the two-way error probability $\\beta\\_{\\leftrightarrow}({{\\left \\vert}\\Psi {\\right \\rangle}})$. Thus, we concentrate ourselves on driving an upper-bound of *β*↔ by constructing a concrete two-way LOCC discrimination protocol. For simplicity, we only treat three-step LOCC discrimination protocols on a bipartite system, which are in the simplest class of genuine two-way LOCC protocols. As a result, we show that even three-step LOCC protocols can discriminates a given state from the completely mixed state much better than by one-way (that is, two-step) LOCC protocols.\nWe can generally describe a three-step LOCC protocol to discriminate a pure state ${{\\left \\vert}\\Psi {\\right \\rangle}}$ from $\\rho \\_{\\rm mix} = \\frac{I}{d^2}$ on a bipartite system H*A* ⊗ H*B* without making any error to detect ${{\\left \\vert}\\Psi {\\right \\rangle}}$ as follow: Suppose there are two parties called Alice and Bob. First, Alice performs a POVM {*M**i*}*i* on her system H*A*, and sends the measurement result *i* to Bob. Second, depending on *i*, Bob performs a POVM {*N**j**i*}*j* on his system H*B*, and sends the measurement result *j* to Alice. If the given state is *ρ*, by easy calculation, we can check that the Alice’s state after this step is\n \n$$\\label{definition of sigma \\_A}\n\\sigma \\_A^{ij} \\stackrel{\\rm def}{=} \n\\frac{\\sqrt{M\\_i}\\sqrt{\\rho \\_A}N\\_j^{iT}\\sqrt{\\rho \\_A}\\sqrt{M\\_i}}{ \\Tr \\left ( M\\_i \\sqrt{\\rho \\_A}N\\_j^{iT}\\sqrt{\\rho \\_A} \\right) },$$ \nwhere $\\rho \\_A \\stackrel{\\rm def}{=} \\Tr \\_B {{\\left \\vert}\\Psi {\\right \\rangle}}{{\\left \\langle}\\Psi {\\right \\vert}}$, and the transposition is taken in the Schmidt basis of ${{\\left \\vert}\\Psi {\\right \\rangle}}$. Thus, in order not to make an error to detect the above state, finally, Alice should make a measurement in {{*σ**A**i**j* > 0}, *I**A* − {*σ**A**i**j* > 0}}, where {*σ**A**i**j* > 0} is a projection operator onto the support of *σ**A**i**j* (the subspace spanned by eigenvectors corresponding to non-zero eigenvalue of *σ**A**i**j* ), *I**A* is an identity operator in H*A*. Then, if she detects {*σ**A**i**j* > 0}, she judges that the first state was ${{\\left \\vert}\\Psi {\\right \\rangle}}$, and if she detects *I**A* − {*σ**A**i**j* > 0}, she judges that the first state was $\\rho \\_{\\rm mix}$. Suppose {*T*, *I**A**B* − *T*} is the POVM corresponds to the above local discrimination protocol, where *T* corresponds to ${{\\left \\vert}\\Psi {\\right \\rangle}}$, and *I* − *T* corresponds to $\\rho \\_{\\rm mix}$ Then, we can check that the whole POVM {*T*, *I* − *T*} can be written down as follows,\n \n$$\\label{definition of two-way T}\nT = \\sum \\_i \\sum \\_j \\left ( \\sqrt{ M\\_i} \\{ \\sigma\\_A^{ij} >0 \\} \\sqrt{ M\\_i} \\right ) \\otimes N\\_j^i,$$ \nwhere *σ**A**i**j* is defined by Eq.([definition of sigma A]), and all *M**i* and *N**j**i* are positive operators satisfying ∑*i**M**i* = *I**A* and ∑*j**N**j**i* = *I**B*. We can also check that *T* defined by Eq.([definition of two-way T]) satisfies ${{\\left \\langle}\\Psi {\\right \\vert}}T{{\\left \\vert}\\Psi {\\right \\rangle}} = 1$ as follows:\n \n$$\\begin{aligned}\n\\fl {{\\left \\langle}\\Psi {\\right \\vert}}T{{\\left \\vert}\\Psi {\\right \\rangle}} &=& \\sum \\_{ij} {{\\left \\langle}\\Psi {\\right \\vert}}\\left ( \\sqrt{M\\_i } \\{ \\sigma\\_A^{ij} >0 \\} \\sqrt{M\\_i } \\right ) \\otimes N\\_j^i{{\\left \\vert}\\Psi {\\right \\rangle}} \\\\\n&=& d \\sum \\_{ij} \\left ( {{\\left \\langle}\\Phi ^+ {\\right \\vert}}\\sqrt{\\rho \\_A} \\otimes I\\_B \\right ) \\left (\\sqrt{M\\_i } \\{ \\sigma\\_A^{ij} >0 \\} \\sqrt{M\\_i } \\right ) \\otimes N\\_j^i\n\\left ( \\sqrt{\\rho \\_A} \\otimes I\\_B {{\\left \\vert}\\Phi^+ {\\right \\rangle}} \\right ) \\\\\n&=& d \\sum \\_{ij} {{\\left \\langle}\\Phi ^+ {\\right \\vert}} \\left ( \\sqrt{N\\_j^i}^T \\sqrt{\\rho \\_A}\\sqrt{M\\_i } \\{ \\sigma\\_A^{ij} >0 \\} \\sqrt{M\\_i } \\sqrt{\\rho \\_A} \\sqrt{N\\_j^i}^T \\right ) \n\\otimes I\\_B {{\\left \\vert}\\Phi ^+ {\\right \\rangle}} \\\\\n&=& \\sum \\_{ij} \\Tr \\sqrt{N\\_j^i}^T \\sqrt{\\rho \\_A}\\sqrt{M\\_i } \\{ \\sigma\\_A^{ij} >0 \\} \\sqrt{M\\_i } \\sqrt{\\rho \\_A} \\sqrt{N\\_j^i}^T \\\\\n&=& \\sum \\_{ij} \\Tr \\sqrt{M\\_i } \\sqrt{\\rho \\_A}N\\_j^{iT}\\sqrt{\\rho \\_A}\\sqrt{M\\_i } \\{ \\sigma\\_A^{ij} >0 \\} \\\\\n&=& \\sum \\_{ij} \\left ( \\Tr M\\_i \\sqrt{\\rho \\_A}N\\_j^{iT}\\sqrt{\\rho \\_A} \\right ) \\cdot \\left ( \\Tr \\sigma \\_A^{ij} \\{ \\sigma\\_A^{ij} >0 \\} \\right )\\\\\n&=& \\sum \\_{ij} \\Tr \\left ( N\\_j^{i}\\left ( \\sqrt{\\rho \\_A} M\\_i \\sqrt{\\rho \\_A} \\right )^T \\right )\\\\\n&=& \\sum \\_{ij} \\Tr \\rho \\_A M \\_i\\\\\n&=& 1,\\end{aligned}$$ \nwhere ${{\\left \\vert}\\Phi ^+ {\\right \\rangle}}$ is the maximally entangled state sharing the Schmidt basis with ${{\\left \\vert}\\Psi {\\right \\rangle}}$, and the transposition *T* is always taken in the Schmidt basis of ${{\\left \\vert}\\Psi {\\right \\rangle}}$. In the second line of the above equalities, we used the equality ${{\\left \\vert}\\Psi {\\right \\rangle}} = \\sqrt{\\rho \\_A} \\otimes I\\_B {{\\left \\vert}\\Phi ^+ {\\right \\rangle}}$. In the third line, we used the equalities $I \\_A \\otimes X {{\\left \\vert}\\Phi ^+ {\\right \\rangle}} = X^T \\otimes I\\_B {{\\left \\vert}\\Phi ^+ {\\right \\rangle}}$, which is valid for an arbitrary operator *X*. In the sixth line, we used Eq.([definition of sigma A]).\nThe above three-step LOCC protocol is enough general. However, it is too complicated to optimize $\\Tr T$ over all choices of POVM {*M**i*}*i* and {*N**j**i*}*j*. In this section, our aim is only to find a good (not necessary optimal) two-way LOCC protocol by which we can discriminate a state from the completely mixed state better than by any one-way LOCC protocols. Thus, to make a problem simpler, we make the following assumptions on Alice’s POVM {*M**i*}*i* and Bob’s POVM {*N**j**i*}*j*: First, without losing any generality, we can write ${{\\left \\vert}\\Psi {\\right \\rangle}}$ as ${{\\left \\vert}\\Psi {\\right \\rangle}} = \\sum \\_{k=1}^d \\sqrt{\\lambda \\_k} {{\\left \\vert}k {\\right \\rangle}}\\otimes {{\\left \\vert}k {\\right \\rangle}}$ with *λ**i* > 0 and *λ**i* ≥ *λ**i* + 1, where $\\{ {{\\left \\vert}k {\\right \\rangle}} \\}\\_{k=1}^d$ is an arbitrary fixing computational basis of H*A* and H*B*, and *d* = dimH*A* = dimH*B*; since our definition of $t\\_{\\leftrightarrow }({{\\left \\vert}\\Psi {\\right \\rangle}})$ (Eq.([definition of tc])) does not depend on the dimension of the whole system, we can always choose the whole system in order that the Schmidt rank of ${{\\left \\vert}\\Psi {\\right \\rangle}}$ is *d*. Second, we assume that the number of POVM element *M**i* is *d*, and *M**i* is diagonalizable in the computational basis $\\{ {{\\left \\vert}k {\\right \\rangle}} \\}\\_{k=1}^d$ as,\n \n$$M\\_i = \\sum \\_{k=1}^{i} \\delta \\_{ki}{{\\left \\vert}k {\\right \\rangle}}{{\\left \\langle}k {\\right \\vert}},$$ \nwhere ${\\rm rank} M\\_i = i$, and the coefficients *δ**k**i* ≥ 0 satisfy ∑*i* = *k**d**δ**k**i* = 1 for all *k*. Moreover, we assume that {*N**j**i*}*j* is a von Neumann measurement corresponding to a mutually unbiased basis $\\{ {{\\left \\vert}\\xi \\_j^i {\\right \\rangle}} \\}\\_{j=1}^{{\\rm rank} M\\_i}$ of an orthonormal set of eigen vectors of $\\omega \\_B \\stackrel{\\rm def}{=} \\frac{\\sqrt{\\rho \\_B}M\\_i^T \\sqrt{\\rho \\_B}}{\\Tr \\sqrt{\\rho \\_B}M\\_i^T \\sqrt{\\rho \\_B}}$ corresponding to non-zero eigenvalues; that is, $\\{ \\xi \\_j^i \\}\\_{j=1}^{{\\rm rank} M\\_i}$ only spans ${\\rm Ran } \\frac{\\sqrt{\\rho \\_B}M\\_i^T \\sqrt{\\rho \\_B}}{\\Tr \\sqrt{\\rho \\_B}M\\_i^T \\sqrt{\\rho \\_B}} $. In other words, an orthonormal set of states $\\{ {{\\left \\vert}\\xi \\_j^i {\\right \\rangle}} \\}\\_{j=1}^{{\\rm rank} M\\_i}$ satisfies\n \n$$\\label{mutually unbiased}\n{{\\left \\langle}\\xi \\_j^i {\\right \\vert}}\\frac{\\sqrt{\\rho \\_B}M\\_i^T \\sqrt{\\rho \\_B}}{\\Tr \\sqrt{\\rho \\_B}M\\_i^T \\sqrt{\\rho \\_B}} \n{{\\left \\vert}\\xi \\_j^i {\\right \\rangle}} = \\frac{1}{{\\rm rank} M\\_i}.$$ \nNote that *ω**B* is the Bob’s state after the Alice’s first measurement in the case where the given state is ${{\\left \\vert}\\Psi {\\right \\rangle}}$, and thus, Bob only needs to detect the subspace ${\\rm Ran} \\omega \\_B$ in this case. That is, Bob’s POVM consists of $\\{ {{\\left \\vert}\\xi \\_j^i {\\right \\rangle}} {{\\left \\langle}\\xi \\_j^i {\\right \\vert}} \\}\\_{j=1}^{{\\rm rank} M\\_i}$ and $I\\_B - \\sum \\_{j=1}^{{\\rm rank } M\\_i} {{\\left \\vert}\\xi \\_j^i {\\right \\rangle}} {{\\left \\langle}\\xi \\_j^i {\\right \\vert}}$; if Bob derives the measurement result corresponding to $I\\_B - \\sum \\_{j=1}^{{\\rm rank } M\\_i} {{\\left \\vert}\\xi \\_j^i {\\right \\rangle}} {{\\left \\langle}\\xi \\_j^i {\\right \\vert}}$, then, he judges that the given state is $\\rho \\_{\\rm mix}$. We also should note that due to this Bob’s mutually unbiased measurement, our three-step protocol can not be reduced to a two-step one-way LOCC protocol. If all Bob’s POVMs are commutative with the eigen basis of *ω**B*, the whole protocol can be reduced to a one-way LOCC protocol; however, *ω**B* never commutes the projection onto the mutually unbiased basis of the eigen basis of *ω**B*. Finally, under the above assumptions, we can write down the trace of the whole POVM element $\\Tr T$ as follows,\n \n$$\\begin{aligned}\n\\fl \\Tr T &=& \\Tr \\left ( \\sum \\_{i=1}^d \\sum \\_{j=1}^{{\\rm rank} M\\_i} \\left ( \\sqrt{M\\_i } \\{ \\sigma \\_A^{ij} > 0\\} \\sqrt{M\\_i } \\right ) \n\\otimes N\\_j^i \\right ) \\nonumber \\\\\n\\fl &=& \\sum \\_{i=1}^d \\sum \\_{j=1}^{{\\rm rank} M\\_i} \\Tr \\left ( \\left ( \\sqrt{M\\_i } \n\\left ( \\frac{\\sqrt{M\\_i}\\sqrt{\\rho \\_A}({{\\left \\vert}\\xi \\_j^i {\\right \\rangle}}{{\\left \\langle}\\xi \\_j^i {\\right \\vert}})^T \\sqrt{\\rho \\_A}\\sqrt{M\\_i}}{{{\\left \\langle}\\xi\\_j^i {\\right \\vert}}\\sqrt{\\rho \\_A}M\\_i^T \\sqrt{\\rho \\_A}{{\\left \\vert}\\xi \\_j^i {\\right \\rangle}}} \n\\right ) \\sqrt{M\\_i } \\right ) \\otimes {{\\left \\vert}\\xi \\_j^i {\\right \\rangle}}{{\\left \\langle}\\xi \\_j^i {\\right \\vert}} \\right ) \\nonumber \\\\\n\\fl &=& \\sum \\_{i=1}^d \\sum \\_{j=1}^{{\\rm rank} M\\_i} \\frac{{{\\left \\langle}\\xi \\_j^i {\\right \\vert}}\\sqrt{\\rho \\_A} (M\\_i^T)^2 \\sqrt{\\rho \\_A} \n{{\\left \\vert}\\xi \\_j^i {\\right \\rangle}}}{{{\\left \\langle}\\xi \\_j^i {\\right \\vert}}\\sqrt{\\rho \\_A} M\\_i^T \\sqrt{\\rho \\_A} {{\\left \\vert}\\xi \\_j^i {\\right \\rangle}}} \\nonumber \\\\\n\\fl &=& \\sum \\_{i=1}^d {\\rm rank } M\\_i \\frac{\\Tr \\sqrt{\\rho \\_A}(M\\_i^T)^2\\sqrt{\\rho \\_A }}{\\Tr \\sqrt{\\rho \\_A}M\\_i^T\\sqrt{\\rho \\_A}} \\nonumber \\\\\n\\fl &=& \\sum \\_{i=1}^d i \\cdot \\frac{\\sum \\_{k=1}^i \\lambda \\_k \\delta \\_{ki}^2}{\\sum \\_{k=1}^i \\lambda \\_i \\delta \\_{ki}}. \\nonumber\\end{aligned}$$ \nIn the second line of the above equalities, we used Eq.([definition of sigma A]) (the definition of *σ**A**i**j* ) and the equality $\\left \\{ \\frac{\\sqrt{M\\_i}\\sqrt{\\rho \\_A}({{\\left \\vert}\\xi \\_j^i {\\right \\rangle}}{{\\left \\langle}\\xi \\_j^i {\\right \\vert}})^T \\sqrt{\\rho \\_A}\\sqrt{M\\_i}}{{{\\left \\langle}\\xi\\_j^i {\\right \\vert}}\\sqrt{\\rho \\_A}M\\_i^T \n\\sqrt{\\rho \\_A}{{\\left \\vert}\\xi \\_j^i {\\right \\rangle}}} \n> 0 \\right \\} = \\frac{\\sqrt{M\\_i}\\sqrt{\\rho \\_A}({{\\left \\vert}\\xi \\_j^i {\\right \\rangle}}{{\\left \\langle}\\xi \\_j^i {\\right \\vert}})^T \n\\sqrt{\\rho \\_A}\\sqrt{M\\_i}}{{{\\left \\langle}\\xi\\_j^i {\\right \\vert}}\\sqrt{\\rho \\_A}M\\_i^T \\sqrt{\\rho \\_A}{{\\left \\vert}\\xi \\_j^i {\\right \\rangle}}} $. In the fourth line of the above equalities, we used the relation *ρ**A* = *ρ**B* and the condition of mutually unbiased basis Eq.([mutually unbiased]). Therefore, our problem is reduced to the optimization of $\\sum \\_{i=1}^d i \\cdot \\frac{\\sum \\_{k=1}^i \\lambda \\_k \\delta \\_{ki}^2}{\\sum \\_{k=1}^i \\lambda \\_i \\delta \\_{ki}}$ over {*δ**k**i*}*k**i* subjected to the constraints *δ**k**i* ≥ 0 and ∑*i* = *k**d**δ**k**i* = 1. In other words, we can summarized the above discussion in the form of the following lemma.\nFor a bipartite pure state ${{\\left \\vert}\\Psi {\\right \\rangle}} \\in {\\mathcal{H}}\\_A \\otimes {\\mathcal{H}}\\_B$, $\\beta\\_{\\leftrightarrow}({{\\left \\vert}\\Psi {\\right \\rangle}})$ satisfies the following inequality,\n \n$$\\begin{aligned}\n\\fl \\beta\\_{\\leftrightarrow}({{\\left \\vert}\\Psi {\\right \\rangle}}) \n&\\le &\\beta\\_{\\widetilde{\\leftrightarrow}}({{\\left \\vert}\\Psi {\\right \\rangle}}) \\nonumber\\\\\n\\fl &\\stackrel{\\rm def}{=} &\n\\frac{1}{d^2}\\min \\_{\\{ \\delta \\_{ki} \\}\\_{1 \\le k \\le i \\le d}} \\left \\{ \n\\sum \\_{i=1}^d i \\cdot \\frac{\\sum \\_{k=1}^i \\lambda \\_k \\delta \\_{ki}^2}{\\sum \\_{k=1}^i \\lambda \\_k \\delta \\_{ki}} \\ \\Big| \\forall k, \\forall i, \n\\delta \\_{ki} \\ge 0, \\ {\\rm and}\\ \\forall k, \\sum \\_{i=k}^d \\delta\\_{ki} =1 \\right \\},\n\\label{upperbound of two-way t}\\end{aligned}$$ \nwhere {*λ**k*}*k* = 1*d* is the Schmidt coefficients of ${{\\left \\vert}\\Psi {\\right \\rangle}}$, and satisfies *λ**k* ≥ *λ**k* + 1 for all *k*, and the indices (*k*, *i*) moves among all of the triangle region 1 ≤ *k* ≤ *i* ≤ *d*.\nThen, the above inequality can write as $\\beta\\_{\\leftrightarrow}({{\\left \\vert}\\Psi {\\right \\rangle}}) \\le \\beta\\_{\\widetilde{\\leftrightarrow}}({{\\left \\vert}\\Psi {\\right \\rangle}})$. Together with the results of the past sections, we derive the following inequalities related to the minimum type 2 error probability for a bipartite pure state:\n \n$$\\begin{aligned}\n\\fl \\beta\\_g({{\\left \\vert}\\Psi {\\right \\rangle}})=\\frac{1}{d^2} \\le \\beta\\_{\\rm sep}({{\\left \\vert}\\Psi {\\right \\rangle}}) && = \\frac{1}{d^2}\n\\Bigl(\\sum \\_{i} \\sqrt{\\lambda \\_i} \\Bigr)^2\n\\nonumber \\\\\n&& \\le \\beta\\_{\\leftrightarrow} ({{\\left \\vert}\\Psi {\\right \\rangle}}) \\le \\beta\\_{\\widetilde{\\leftrightarrow}}({{\\left \\vert}\\Psi {\\right \\rangle}})\n\\le \\frac{1}{d^2}{\\rm rank}\\Tr \\_B({{\\left \\vert}\\Psi {\\right \\rangle}}{{\\left \\langle}\\Psi {\\right \\vert}}) = \\beta\\_{\\rightarrow}({{\\left \\vert}\\Psi {\\right \\rangle}}). \\label{all inequalities}\\end{aligned}$$ \nFor a two-qubit system, we can analytically calculate the exact value of the upper-bound $\\beta\\_{\\widetilde{\\leftrightarrow}}({{\\left \\vert}\\Psi {\\right \\rangle}})$, and can derive the following lemma.\nIn a two-qubit system,\n \n$$\\label{two-way bound}\n\\beta\\_{\\widetilde{\\leftrightarrow} } ({{\\left \\vert}\\Psi {\\right \\rangle}} ) = \\frac{1}{2} - \\frac{(1-\\sqrt{2\n\\lambda })^2}{4 \\left (1- \\lambda \\right ) },$$ \nwhere {1 − *λ*, *λ*} is the Schmidt coefficient of ${{\\left \\vert}\\Psi {\\right \\rangle}}$ satisfying $1 \\le \\lambda \\le \\frac{1}{2}$.\nWithout losing generality, we can write a bipartite state as ${{\\left \\vert}\\Psi {\\right \\rangle}} = \\sqrt{1-\\lambda} {{\\left \\vert}00 {\\right \\rangle}} + \\sqrt{\\lambda} {{\\left \\vert}11 {\\right \\rangle}}$. Then, by a straightforward calculation, we derive\n \n$$\\beta\\_{\\widetilde{\\leftrightarrow}}({{\\left \\vert}\\Psi {\\right \\rangle}}) = \\frac{1}{4}\\min\\_{0 \\le \\delta \\le 1} \n\\left \\{ 2 -\\frac{\\delta \\left \\{ (1-2\\lambda) + \\delta (1-\\lambda) \\right \\} }{1-\\delta (1-\\lambda )} \\right \\},$$ \nwhere we substitute *λ*1 = 1 − *λ*, *λ*2 = *λ*, *δ*11 = *δ*, *δ*12 = 1 − *δ*, and *δ*22 = 1 into Eq.([upperbound of two-way t]). Suppose $t'\\_{\\widetilde{\\leftrightarrow}}(\\lambda, \\delta ) \n\\stackrel{\\rm def}{=} 2 -\\frac{\\delta \\left \\{ (1-2\\lambda) + \\delta (1-\\lambda) \\right \\} }{1-\\delta (1-\\lambda )}$. Then, we can calculate the derivative of $t'\\_{\\widetilde{\\leftrightarrow}}(\\lambda, \\delta )$ as\n \n$$\\frac{ \\partial t'\\_{\\widetilde{\\leftrightarrow}}(\\lambda, \\delta )}{\\partial \\delta } = - \\{\n(1-\\lambda )\\delta - (1-\\sqrt{2\\lambda}) \\} \\{ (1-\\lambda )\\delta\n- (1+ \\sqrt{2\\lambda })\\},$$ \nfor fixed $0 \\le \\lambda \\le \\frac{1}{2} $. Thus, under the condition 0 ≤ *δ* ≤ 1, $t'\\_{\\widetilde{\\leftrightarrow}}(\\lambda,\n\\delta )$ attains its minimum when $\\delta =\n\\frac{1-\\sqrt{2\\lambda}}{1-\\lambda}$. Therefore, we derive\n \n$$\\begin{aligned}\n \\beta\\_{\\widetilde{\\leftrightarrow}}({{\\left \\vert}\\Psi {\\right \\rangle}})\n&=& \\frac{1}{4}\\min \\_{0 \\le \\delta \\le 1} t'\\_{\\widetilde{\\leftrightarrow}}(\\lambda, \\delta ) \n = \\frac{1}{2} -\n\\frac{(1-\\sqrt{2\\lambda })^2}{4\\left (1-\\lambda \\right )}.\\end{aligned}$$ \n▫\nTherefore, for a two-qubit state ${{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}} = \\sqrt{1-\\lambda}{{\\left \\vert}00 {\\right \\rangle}}+\\sqrt{\\lambda}{{\\left \\vert}11 {\\right \\rangle}}$, the inequality ([all inequalities]) can be reduced as follows,\n \n$$\\begin{aligned}\n\\fl \\beta\\_g({{\\left \\vert}\\Psi {\\right \\rangle}})=\\frac{1}{4} \\le \\beta\\_{\\rm sep} ({{\\left \\vert}\\Psi {\\right \\rangle}} ) =\\frac{1}{4}+ \\frac{1}{2} \\sqrt{\\lambda (1-\\lambda)} \\le \\beta\\_{\\leftrightarrow } ({{\\left \\vert}\\Psi {\\right \\rangle}} )\n&& \\le \\frac{1}{2} - \\frac{(1-\\sqrt{2\n\\lambda })^2}{4\\left ( 1- \\lambda \\right )} \\\\\n\\fl && \\le \\frac{1}{2} = \\beta\\_{\\rightarrow} ({{\\left \\vert}\\Psi {\\right \\rangle}} ),\\end{aligned}$$ \nwhere the equality of the last inequality holds, if and only if the state is a product state or a maximally entangled state. We present the graph of these bounds in **Figure.1**.\n[graph]\nThe bound as a function of *λ* (the Schmidt coefficient of ${{\\left \\vert}\\Psi {\\right \\rangle}}$ ). The thin line: $\\beta\\_{\\rm sep} ({{\\left \\vert}\\Psi {\\right \\rangle}} )$, the broken line: $\\beta\\_{\\widetilde{\\leftrightarrow}}({{\\left \\vert}\\Psi {\\right \\rangle}})$ (an upper bound of $\\beta\\_{\\leftrightarrow}({{\\left \\vert}\\Psi {\\right \\rangle}})$), the thick line: $\\beta\\_{\\rightarrow} ({{\\left \\vert}\\Psi {\\right \\rangle}} )$, the thin broken line: $\\beta\\_{g} ({{\\left \\vert}\\Psi {\\right \\rangle}} )$.\nFrom this figure, we can see that there is a big gap between $\\beta\\_{\\rightarrow}({{\\left \\vert}\\Psi {\\right \\rangle}})$ and $\\beta\\_{\\leftrightarrow}({{\\left \\vert}\\Psi {\\right \\rangle}})$ and the difference between $\\beta\\_{\\rightarrow } ({{\\left \\vert}\\Psi {\\right \\rangle}}) $ and $\\beta\\_{\\rm sep}({{\\left \\vert}\\Psi {\\right \\rangle}})$ is (if the difference exists) relatively small. Thus, for any non-maximally entangled pure states, there is a gap between the one-way and two-way local distinguishability at least for two-qubit systems in terms of $\\beta\\_{\\rightarrow(\\leftrightarrow\n)}({{\\left \\vert}\\Psi {\\right \\rangle}})$.\nIn a system with a dimension of local systems *d* ≥ 3, the optimization in the definition of $\\beta\\_{\\widetilde{\\leftrightarrow}}({{\\left \\vert}\\Psi {\\right \\rangle}})$ (Eq.([upperbound of two-way t])) is too complicated to be solved by an analytical calculation, anymore [1](#fn1). Thus, we numerically calculate the right hand side of Eq.([upperbound of two-way t]) for a C3 ⊗ C3 (two-qutrit) system and a C4 ⊗ C4 system. For a C3 ⊗ C3 system, we calculate Eq.([upperbound of two-way t]) for three different one-parameter families of pure states:\n1. ${{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}} = \\sqrt{1-2\\lambda} {{\\left \\vert}11 {\\right \\rangle}} + \\sqrt{\\lambda} {{\\left \\vert}22 {\\right \\rangle}} + \\sqrt{\\lambda} {{\\left \\vert}33 {\\right \\rangle}}, \n\\ ( 0 \\le \\lambda \\le \\frac{1}{3})$: In this case, $\\beta\\_g({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}})=\\frac{1}{9}$, $\\beta\\_{\\rm sep}({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}}) = \\frac{1}{9}(\\sqrt{1-2\\lambda} + 2\\sqrt{\\lambda})^2$ and $\\beta\\_{\\rightarrow} ({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}})=\\frac{1}{3}$. We give the results of numerical calculation of $\\beta\\_{\\widetilde{\\rm sep}} ({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}} ) $ in **Figure.2**.\n[image31]\nThe bound as a function of *λ* for a family of states ${{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}} = \\sqrt{1-2\\lambda} {{\\left \\vert}11 {\\right \\rangle}} + \\sqrt{\\lambda} {{\\left \\vert}22 {\\right \\rangle}} + \\sqrt{\\lambda} {{\\left \\vert}33 {\\right \\rangle}}$. The thick broken line: results of a numerical calculation of $\\beta\\_{\\widetilde{\\leftrightarrow}} ({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}} ) $ (a lower bound of $\\beta\\_{\\leftrightarrow}({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}})$, the thin line: $\\beta\\_{\\rm sep}({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}})$, the thick line: $\n\\beta\\_{\\rightarrow} ({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}} )$: the thin broken line: $\\beta\\_g({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}})$.\n2. ${{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}} = \\sqrt{1-3\\lambda} {{\\left \\vert}11 {\\right \\rangle}} + \\sqrt{2\\lambda} {{\\left \\vert}22 {\\right \\rangle}} + \\sqrt{\\lambda} {{\\left \\vert}33 {\\right \\rangle}}, \\ ( 0 \\le \\lambda \\le \\frac{1}{5})$: In this case, $\\beta\\_g({{\\left \\vert}\\Psi \\_{\\lambda } {\\right \\rangle}})=\\frac{1}{9}$, $\\beta\\_{\\rm sep}({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}}) = \\frac{1}{9}(\\sqrt{1-3\\lambda} + (1+\\sqrt{2})\\sqrt{\\lambda})^2$ and $\\beta\\_{\\rightarrow} ({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}} )=\\frac{1}{3}$. We give the results of a numerical calculation of $\\beta\\_{\\widetilde{\\rm sep}} ({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}} ) $ in **Figure.3**\n[image32]\nThe bound as a function of *λ* for a family of states ${{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}} = \\sqrt{1-3\\lambda} {{\\left \\vert}11 {\\right \\rangle}} + \\sqrt{2\\lambda} {{\\left \\vert}22 {\\right \\rangle}} + \\sqrt{\\lambda} {{\\left \\vert}33 {\\right \\rangle}}$. The thick broken line: results of a numerical calculation of $\\beta\\_{\\widetilde{\\leftrightarrow}} ({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}} ) $ (a lower bound of $\\beta\\_{\\leftrightarrow}({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}})$, the thin line: $\\beta\\_{\\rm sep}({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}})$, the thick line: $\n\\beta\\_{\\rightarrow} ({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}} )$, the thin broken line: $\\beta\\_g({{\\left \\vert}\\Psi \\_{\\lambda } {\\right \\rangle}})$.\n3. ${{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}} = \\sqrt{1-4\\lambda} {{\\left \\vert}11 {\\right \\rangle}} + \\sqrt{3\\lambda} {{\\left \\vert}22 {\\right \\rangle}} + \\sqrt{\\lambda} {{\\left \\vert}33 {\\right \\rangle}}, \\ ( 0 \\le \\lambda \\le \\frac{1}{7})$: In this case, $\\beta\\_g({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}})=\\frac{1}{9}$, $\\beta\\_{\\rm sep}({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}}) = \\frac{1}{9}(\\sqrt{1-4\\lambda} + (1+\\sqrt{3})\\sqrt{\\lambda})^2$ and $\\beta\\_{\\rightarrow} ({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}} )=\\frac{1}{3}$. We give the results of a numerical calculation of $\\beta\\_{\\widetilde{\\rm sep}} ({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}} ) $ in **Figure.4**\n[image33]\nThe bound as a function of *λ* for a family of states ${{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}}=\\sqrt{1-4\\lambda} {{\\left \\vert}11 {\\right \\rangle}} + \\sqrt{3\\lambda} {{\\left \\vert}22 {\\right \\rangle}} + \\sqrt{\\lambda} {{\\left \\vert}33 {\\right \\rangle}}$. The thick broken line: results of a numerical calculation of $\\beta\\_{\\widetilde{\\leftrightarrow}} ({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}} ) $ (a lower bound of $\\beta\\_{\\leftrightarrow}({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}})$, the thin line: $\\beta\\_{\\rm sep}({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}})$, the thick line: $\n\\beta\\_{\\rightarrow} ({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}} )$, the thin broken line: $\\beta\\_g({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}})$.\nFrom **Figures 2, 3, and 4**, we can confirm that the shapes of the graphs of $\\beta\\_{\\rm sep}({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}})$, and $\\beta\\_{\\widetilde{\\leftrightarrow}}({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}})$ hardly depend of the choice of a one-parameter family ${{\\left \\vert}\\Psi \\_{\\lambda } {\\right \\rangle}}$ in C3 ⊗ C3. For a C4 ⊗ C4 system, we calculate Eq.([upperbound of two-way t]) for two different one-parameter families of pure states:\n1. ${{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}} = \\sqrt{1-3\\lambda} {{\\left \\vert}11 {\\right \\rangle}} + \\sqrt{\\lambda} ({{\\left \\vert}22 {\\right \\rangle}} + {{\\left \\vert}33 {\\right \\rangle}}+ {{\\left \\vert}44 {\\right \\rangle}}), \n\\ ( 0 \\le \\lambda \\le \\frac{1}{4})$: In this case, $\\beta\\_g({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}})=\\frac{1}{16}$, $\\beta\\_{\\rm sep}({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}}) = \\frac{1}{16}(\\sqrt{1-3\\lambda} + 3\\sqrt{\\lambda})^2$ and $\\beta\\_{\\rightarrow} ({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}})=\\frac{1}{4}$. We give the results of numerical calculation of $\\beta\\_{\\widetilde{\\rm sep}} ({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}} ) $ in **Figure.5**.\n[image41]\nThe bound as a function of *λ* for a family of states ${{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}} = \\sqrt{1-3\\lambda} {{\\left \\vert}11 {\\right \\rangle}} + \\sqrt{\\lambda} ({{\\left \\vert}22 {\\right \\rangle}} + {{\\left \\vert}33 {\\right \\rangle}}+ {{\\left \\vert}44 {\\right \\rangle}})$. The thick broken line: results of a numerical calculation of $\\beta\\_{\\widetilde{\\leftrightarrow}} ({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}} ) $ (a lower bound of $\\beta\\_{\\leftrightarrow}({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}})$, the thin line: $\\beta\\_{\\rm sep}({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}})$, the thick line: $\n\\beta\\_{\\rightarrow} ({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}} )$, the thin broken line: $\\beta\\_g({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}})$.\n2. ${{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}} = \\sqrt{1-\\frac{9}{2}\\lambda} {{\\left \\vert}11 {\\right \\rangle}} + \\sqrt{2\\lambda} {{\\left \\vert}22 {\\right \\rangle}} + \n\\sqrt{\\frac{3}{2}\\lambda} {{\\left \\vert}33 {\\right \\rangle}} + \\sqrt{\\lambda}{{\\left \\vert}44 {\\right \\rangle}}, \\ ( 0 \\le \\lambda \\le \\frac{2}{13})$: In this case, $\\beta\\_g({{\\left \\vert}\\Psi \\_{\\lambda } {\\right \\rangle}})=\\frac{1}{16}$, $\\beta\\_{\\rm sep}({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}}) = \n\\frac{1}{16} \\left (\\sqrt{1-\\frac{9}{2}\\lambda} + \\left (1+\\sqrt{\\frac{3}{2}}+\\sqrt{2} \\right )\\sqrt{\\lambda} \\right )^2$ and $\\beta\\_{\\rightarrow} ({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}} )=\\frac{1}{4}$. We give the results of numerical calculation of $\\beta\\_{\\widetilde{\\rm sep}} ({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}} ) $ in **Figure.6**\n[image42]\nThe bound as a function of *λ* for a family of states ${{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}} = {{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}} = \\sqrt{1-\\frac{9}{2}\\lambda} {{\\left \\vert}11 {\\right \\rangle}} + \\sqrt{2\\lambda} {{\\left \\vert}22 {\\right \\rangle}} + \n\\sqrt{\\frac{3}{2}\\lambda} {{\\left \\vert}33 {\\right \\rangle}} + \\sqrt{\\lambda}{{\\left \\vert}44 {\\right \\rangle}}$. The thick broken line: results of a numerical calculation of $\\beta\\_{\\widetilde{\\leftrightarrow}} ({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}} ) $ (a lower bound of $\\beta\\_{\\leftrightarrow}({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}})$, the thin line: $\\beta\\_{\\rm sep}({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}})$, the thick line: $\n\\beta\\_{\\rightarrow} ({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}} )$, the thin broken line: $\\beta \\_g ({{\\left \\vert}\\Psi \\_{\\lambda } {\\right \\rangle}})$.\nFrom **Figures. 5, and 6**, we can confirm that the shapes of the graphs of $\\beta\\_{\\rm sep}({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}})$, and $\\beta\\_{\\widetilde{\\leftrightarrow}}({{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}})$ hardly depend of the choice of a one-parameter family ${{\\left \\vert}\\Psi \\_{\\lambda } {\\right \\rangle}}$ in C4 ⊗ C4 as well as in C3 ⊗ C3. Note that, for the all above families of states, we choose a parameter *λ* so that ${{\\left \\vert}\\Psi \\_{\\lambda} {\\right \\rangle}}$ can be converted to ${{\\left \\vert}\\Psi \\_{\\lambda'} {\\right \\rangle}}$ by LOCC for all *λ* ≥ *λ*ʹ, and ${{\\left \\vert}\\Psi \\_0 {\\right \\rangle}}$ is a product state; that is, in a naive sense, the degree of entanglement increases monotonically when *λ* increases. From **Figures 2,3,4,5,6**, as well as for a two-qubit system (**Figure.1**), we can see that there is always a big gap between $\\beta\\_{\\rightarrow}({{\\left \\vert}\\Psi {\\right \\rangle}})$ and $\\beta\\_{\\leftrightarrow}({{\\left \\vert}\\Psi {\\right \\rangle}})$ and the difference between $\\beta\\_{\\rightarrow } ({{\\left \\vert}\\Psi {\\right \\rangle}}) $ and $\\beta\\_{\\rm sep}({{\\left \\vert}\\Psi {\\right \\rangle}})$ is (if the difference exists) relatively small for C3 ⊗ C3 and C4 ⊗ C4 systems. Moreover, since the shape of graph corresponding to $\\beta\\_{\\widetilde{\\leftrightarrow}}({{\\left \\vert}\\Psi {\\right \\rangle}})$ seems not to change depending on a dimension of a system, we may guess that, for any non-maximally entangled pure states (even in a high dimensional system), there is a gap between the one-way and two-way local distinguishability in terms of $\\beta\\_{\\rightarrow(\\leftrightarrow\n)}({{\\left \\vert}\\Psi {\\right \\rangle}})$. That is, the two-way classical communication remarkably improves the local distinguishability compared to the local discrimination by the one-way classical communication at least for bipartite pure states.\nConclusion\n==========\nIn this paper, in order to clarify the difference of the two-way LOCC and the one-way LOCC on local discrimination problems, we concentrated ourselves on the local discrimination of a given bipartite state from the completely mixed state $\\rho\\_{\\rm mix}$ under the condition where the given state should be detected perfectly while the previous researches treated the same problem between two bipartite pure states. We defined *β*→(*ρ*), *β*↔(*ρ*), and $\\beta\\_{\\rm sep}(\\rho)$ as the minimum error probability to detect the completely mixed state by the one-way LOCC, the two-way LOCC, and the separable operation, respectively, under the condition that a given state *ρ* is detected perfectly. First, in Section [separable section chap.3], for separable operations, we showed that the minimum error probability $t\\_{\\rm sep} (\\rho)$ coincides with an entanglement measure called the global robustness of entanglement for a bipartite pure state except an unimportant constant term. Then, in Section [bipartite], for one-way LOCC, we showed that the minimum error probability *β*→(*ρ*) coincides with the Schmidt rank for a bipartite pure state except an unimportant constant term. Finally, in Section [two-way], by constructing a concrete three-step two-way LOCC discrimination protocol, we derived an upper bound for the minimum error probability *β*↔(*ρ*) for a bipartite pure state. By calculating this upper bound analytically and also numerically, we showed that *β*↔(*ρ*) is strictly smaller than *β*→(*ρ*), and moreover, *β*↔(*ρ*) and $\\beta\\_{\\rm sep}(\\rho)$ give almost the same value for a lower dimensional bipartite pure state; this results can be seen in Figures 2,3,4,5,6. As a result, although there is no difference between the one-way LOCC and the two-way LOCC concerning local discrimination between two bipartite pure states, we conclude that the two-way classical communication remarkably improves the local distinguishability in comparison with the one-way classical communication for a low-dimensional pure state at least in the present problem setting. Due to our quantitative comparison, from the continuity of the second kinds of error probabilities, a similar result should holds when the second state *ρ̃* belongs to the neighborhood of the completely mixed state. Further, we are preparing a forthcoming manuscript concerning this kind of problem in the case of multi-partite case in the near future.\nAcknowledgments\n===============\nMO would like to thank Prof. Mio Murao, Dr. Damian Markham, and Dr. Shashank Virmani for helpful comments and discussions. MO is grateful to Professor Hiroshi Imai for his support and encouragement The authors also appreciate Reviewr’s comments.\nThis work was supported by Special Coordination Funds for Promoting Science and Technology and a MEXT Grant-in-Aid for Scientific Research on Priority, Deepening and Expansion of Statistical Mechanical Informatics (DEX-SMI), No. 18079014.\nReferences\n==========\n99 C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher *Phys. Rev. A* **53**, 2046-2052 (1996); M. A. Nielsen, *Phys. Rev. Lett.* **83** 436 (1999); G. Vidal, *Phys. Rev. Lett.* **83**, 1046 (1999). H. K. Lo, and S. Popescu, *Phys. Rev. A*, 63, 022301 (2001) M. Hayashi, *Quantum Information: An Introduction*, Springer-Verlag, (2006) C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, *Phys. Rev. A* **54**, 3824 (1996) C. H. Bennett, D. P. DiVincenzo, C. A. Fuchs, T. Mor, E. Rains, P. W. Shor, J. A. Smolin, and W. K. Wootters *Phys. Rev. A* **59**, 1070-1091 (1999); S. Ghosh, G. Kar, A. Roy, and D. Sarkar, *Phys. Rev. A* **70**, 022304 (2004) ; H. Fan, *Phys. Rev. Lett.* **92**, 177905 (2004); M. Nathanson, *J. Math. Phys.* 46, 062103 (2005) B. Groisman and L. Vaidman, *J. Phys. A*, **34**, 6881 (2001). S. M. Cohen, *Phy. Rev. A*, **75**, 052313 (2007). A. S. Holevo, *IEEE Trans. Infor. Theory*, **44**, 269 (1998). M. V. Burnashev and A. S. Holevo, *Prob. of Inform. Trans.*, **34**, 97 (1998). B. Schumacher and M. D. Westmoreland, *Phys. Rev. A*, **56**, 131, (1997). M. Hayashi and H. Nagaoka: *IEEE Trans. Infor. Theory*, **49**, 1753 (2003). T. Ogawa and H. Nagaoka, *IEEE Trans. Infor. Theory*, **53**, 2261 - 2266 (2007). A. Winter. *quant-ph/*0507045. A. Ambainis, *quant-ph/*9902053. M. Hayashi, A. Kawachi, and H. Kobayashi, *quant-ph/*0604174 *to appear in Quantum Information and Computation*. S. Virmani, M. Sacchi, M. B. Plenio and D. Markham, *Phys. Lett. A* **288**, 62 (2001)\nJ. Walgate, A. J. Short, L. Hardy and V. Vedral, *Phys. Rev. Lett.* **85**, 4972 (2000) G. Alagic, C. Moore, and A. Russell, *quant-ph/*0511054, (2005); M. Grigni, L. J. Schulman, M. Vazirani, and U. Vazirani, *Combinatorica*, **24**, 137, (2004); S. Hallgren, *Proceedings of the 34th Annual ACM Symposium on Theory of Computing*, 653, (2002); S. Hallgren, *Proceedings of the 37th Annual ACM Symposium on Theory of Computing*, 468, (2005); J. Kempe and A. Shalev, *Proceedings of the 16th ACM-SIAM Symposium on Discrete Algorithms*, 1118, (2005); C. Moore, D. Rockmore, A. Russell, and L. J. Schulman. *Proceedings of the 15th ACM-SIAM Symposium on Discrete Algorithms*, 1106, (2004); C. Moore and A. Russell, *quant-ph/*0501177, (2005).\nLehmann, E. L. 1986 *Testing statistical hypotheses*, Second edition, Wiley.\nM. Hayashi, D. Markham, M. Murao, M. Owari, and S. Virmani, *Phys. Rev. Lett.* **96**, 040501 (2006) E. M. Rains, *IEEE Trans. Info. Th.* **47**, 2921, (2001); M. J. Donald, M. Horodecki, O. Rudolph, *J. Math. Phys.* **43** 4252 (2002) K.M.R. Audenaert, J. Calsamiglia, Ll. Masanes, R. Munoz-Tapia, A. Acin, E. Bagan, and F. Verstraete, *Phys. Rev. Lett.* **98**, 160501 (2007). M. Nussbaum and A. Szkola, “A lower bound of Chernoff type for symmetric quantum hypothesis testing,” quant-ph/0607216. S. Massar and S. Popescu, “Optimal Extraction of Information from Finite Quantum Ensembles,” *Phys. Rev. Lett.*, **74**, 1259 (1995). J. A. Bucklew, *Large Deviation Techniques in Decision, Simulation, and Estimation*, (John Wiley & Sons, 1990).\nW. Hoeffding, “Asymptotically optimal test for multinomial distributions,” *Ann. Math. Stat.*, **36**, 369-400 (1965).\nF. Hiai and D. Petz, “The proper formula for relative entropy and its asymptotics in quantum probability,” *Comm. Math. Phys.*, **143**, 99–114, (1991).\nT. Ogawa and H. Nagaoka, “Strong converse and Stein’s lemma in quantum hypothesis testing,” *IEEE Trans. Infor. Theory*, **46**, 2428–2433 (2000); quant-ph/9906090 (1999).\nM. Hayashi, “Error Exponent in Asymmetric Quantum Hypothesis Testing and Its Application to Classical-Quantum Channel coding,” quant-ph/0611013; to appear in *Phys. Rev. A*.\nH. Nagaoka, “The Converse Part of The Theorem for Quantum Hoeffding Bound,” quant-ph/0611289.\nG. Vidal, R. Tarrach, *Phys. Rev. A* **59** (1999) 141-155 C. H. Bennett, D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin, B. M. Terhal, *Phys.Rev.Lett.* **82**, 5385, (1999) private communication with S. Virmani and D. Markham M. Hayashi, D. Markham, M. Murao, M. Owari, and S. Virmani, arXiv:0710.1056. “*Entanglement and group symmetries: stabilizer, symmetric and antisymmetric states*”. E. M. Rains, *IEEE Trans. Inf. Theory*, **47**, 2921 (2001); T. Hiroshima and M. Hayashi, *Phys. Rev. A*, **70**, 030302(R) (2004) W. K. Wootters and B. C. Fields, *Ann. Phys.*, **191**, 363 (1989). M. Owari and M. Hayashi, *in preparation*.\nProof of Eq.([average])\n=======================\nNow, we prove Eq.([average]), which is used in proof of Theorem [theorem separable main]. Suppose $P \\stackrel{\\rm def}{=} \\frac{1}{2}\\sum \\_{i\\neq j}\n{{\\left \\vert}\\overline{a\\_{ij}} {\\right \\rangle}}{{\\left \\langle}\\overline{a\\_{ij}} {\\right \\vert}} \n\\otimes {{\\left \\vert}\\overline{b\\_{ij}} {\\right \\rangle}}{{\\left \\langle}\\overline{b\\_{ij}} {\\right \\vert}}$ and $Q \\stackrel{\\rm def}{=} \\sum \\_{i\\neq j}\\{ \\sum \\_{k \\neq i,j} \\lambda \\_k + (\\sqrt{\\lambda \\_i} - \\sqrt{\\lambda \\_j})^2\\}{{\\left \\vert}e\\_if\\_j {\\right \\rangle}} {{\\left \\langle}e\\_if\\_j {\\right \\vert}}$; that is, $\\overline{T\\_0} = P + Q$. Then, by applying twirling operation over *U**θ⃗*, we drive the following equality:\n \n$$\\begin{aligned}\n\\fl &\\quad & \\int \\_0^{2 \\pi} \\cdots \\int \\_0^{2\n\\pi} U\\_{\\overrightarrow{\\theta}} P U\\_{\\overrightarrow{\\theta}}^{\\dagger} d \\theta \\_1 \\cdots d \\theta \\_d \\nonumber \\\\\n\\fl &=& \n(\\sum \\_{j'=1}^d {{\\left \\vert}e\\_{j'} {\\right \\rangle}}{{\\left \\langle}e\\_{j'} {\\right \\vert}} \\otimes {{\\left \\vert}f\\_{j'} {\\right \\rangle}}{{\\left \\langle}f\\_{j'} {\\right \\vert}} ) \nP\n(\\sum \\_{j'=1}^d {{\\left \\vert}e\\_{j'} {\\right \\rangle}}{{\\left \\langle}e\\_{j'} {\\right \\vert}} \\otimes {{\\left \\vert}f\\_{j'} {\\right \\rangle}}{{\\left \\langle}f\\_{j'} {\\right \\vert}} ) \n\\\\\n\\fl &\\quad & +\\sum \\_{i' \\neq k'} \n\\left( {{\\left \\vert}e\\_{i'} {\\right \\rangle}}{{\\left \\langle}e\\_{i'} {\\right \\vert}}\\otimes {{\\left \\vert}f\\_{k'} {\\right \\rangle}}{{\\left \\langle}f\\_{k'} {\\right \\vert}} \\right ) \nP \n\\left( {{\\left \\vert}e\\_{i'} {\\right \\rangle}}{{\\left \\langle}e\\_{i'} {\\right \\vert}}\\otimes {{\\left \\vert}f\\_{k'} {\\right \\rangle}}{{\\left \\langle}f\\_{k'} {\\right \\vert}} \\right ) \\end{aligned}$$ \nThis equality can be proven as follows: The action of a twirling operation (group-averaging) over a unitary representation of a compact topological group is equal to the action of the projection onto the subspace of all invariant elements under the group action. For the action of *U**θ⃗* and *U**θ⃗*†, the subspace (of operator-space B(H)) consisting of all the invariant element is spanned by the operators $\\{ {{\\left \\vert}e\\_jf\\_k {\\right \\rangle}}{{\\left \\langle}e\\_jf\\_k {\\right \\vert}} \\}\\_{j \\neq k}$ and $\\{ {{\\left \\vert}e\\_jf\\_j {\\right \\rangle}}{{\\left \\langle}e\\_kf\\_k {\\right \\vert}} \\}\\_{ij}$. Therefore, we can easily see the above equation. For *i* ≠ *j*, *i*ʹ ≠ *k*ʹ, we have\n \n$$\\begin{aligned}\n\\fl (\\sum \\_{j'=1}^d {{\\left \\vert}e\\_{j'} {\\right \\rangle}}{{\\left \\langle}e\\_{j'} {\\right \\vert}} \\otimes {{\\left \\vert}f\\_{j'} {\\right \\rangle}}{{\\left \\langle}f\\_{j'} {\\right \\vert}} ) \n{{\\left \\vert}\\overline{a\\_{ij}} {\\right \\rangle}} {{\\left \\vert}\\overline{b\\_{ij}} {\\right \\rangle}}\n&=& \\sqrt{\\lambda\\_j}{{\\left \\vert}e\\_i f\\_i {\\right \\rangle}} -\\sqrt{\\lambda\\_i}{{\\left \\vert}e\\_j f\\_j {\\right \\rangle}} \\\\\n\\fl \\left( {{\\left \\vert}e\\_{i'} {\\right \\rangle}}{{\\left \\langle}e\\_{i'} {\\right \\vert}}\\otimes {{\\left \\vert}f\\_{k'} {\\right \\rangle}}{{\\left \\langle}f\\_{k'} {\\right \\vert}} \\right ) \n{{\\left \\vert}\\overline{a\\_{ij}} {\\right \\rangle}} {{\\left \\vert}\\overline{b\\_{ij}} {\\right \\rangle}}\n&=&\n\\delta\\_{i',i}\\delta\\_{k',j}\n(\\lambda\\_j\\lambda\\_j)^{\\frac{1}{4}}{{\\left \\vert}e\\_i f\\_j {\\right \\rangle}} \n-\n\\delta\\_{i',j}\\delta\\_{k',i}\n(\\lambda\\_j\\lambda\\_j)^{\\frac{1}{4}}{{\\left \\vert}e\\_j f\\_i {\\right \\rangle}}.\\end{aligned}$$ \nSince\n \n$$\\begin{aligned}\n\\fl &&(\\sqrt{\\lambda\\_j}{{\\left \\vert}e\\_i f\\_i {\\right \\rangle}} -\\sqrt{\\lambda\\_i}{{\\left \\vert}e\\_j f\\_j {\\right \\rangle}})\n(\\sqrt{\\lambda\\_j}{{\\left \\langle}e\\_i f\\_i {\\right \\vert}} -\\sqrt{\\lambda\\_i}{{\\left \\langle}e\\_j f\\_j {\\right \\vert}})\\\\\n\\fl &=&\n\\lambda\\_j{{\\left \\vert}e\\_i f\\_i {\\right \\rangle}}{{\\left \\langle}e\\_i f\\_i {\\right \\vert}}\n+\\lambda\\_i{{\\left \\vert}e\\_j f\\_j {\\right \\rangle}}{{\\left \\langle}e\\_j f\\_j {\\right \\vert}}\n- \\sqrt{\\lambda\\_j}\\sqrt{\\lambda\\_i}\n{{\\left \\vert}e\\_i f\\_i {\\right \\rangle}}{{\\left \\langle}e\\_j f\\_j {\\right \\vert}}\n- \\sqrt{\\lambda\\_i}\\sqrt{\\lambda\\_j}\n{{\\left \\vert}e\\_j f\\_j {\\right \\rangle}}{{\\left \\langle}e\\_i f\\_i {\\right \\vert}},\\end{aligned}$$ \nwe obtain\n \n$$\\begin{aligned}\n\\fl &\\quad & \\int \\_0^{2 \\pi} \\cdots \\int \\_0^{2\n\\pi} U\\_{\\overrightarrow{\\theta}} P U\\_{\\overrightarrow{\\theta}}^{\\dagger} d \\theta \\_1 \\cdots d \\theta \\_d \\nonumber \\\\\n\\fl &=& \n\\sum\\_{i \\neq j}\n\\Bigl(\\lambda\\_j{{\\left \\vert}e\\_i f\\_i {\\right \\rangle}}{{\\left \\langle}e\\_i f\\_i {\\right \\vert}}\n+\\lambda\\_i{{\\left \\vert}e\\_j f\\_j {\\right \\rangle}}{{\\left \\langle}e\\_j f\\_j {\\right \\vert}}\n- \\sqrt{\\lambda\\_j}\\sqrt{\\lambda\\_i}\n{{\\left \\vert}e\\_i f\\_i {\\right \\rangle}}{{\\left \\langle}e\\_j f\\_j {\\right \\vert}}\n- \\sqrt{\\lambda\\_i}\\sqrt{\\lambda\\_j}\n{{\\left \\vert}e\\_j f\\_j {\\right \\rangle}}{{\\left \\langle}e\\_i f\\_i {\\right \\vert}}\\Bigr)\\\\\n\\fl &&\n+ \\sum \\_{i\\neq j} \\sqrt{\\lambda \\_i\\lambda \\_j}{{\\left \\vert}e\\_if\\_j {\\right \\rangle}}{{\\left \\langle}e\\_if\\_j {\\right \\vert}} \n\\\\\n\\fl &=& (\\sum \\_i^d {{\\left \\vert}e\\_if\\_i {\\right \\rangle}}{{\\left \\langle}e\\_if\\_i {\\right \\vert}}) - (\\sum \\_{i=1}^d \\sqrt{\\lambda \\_i} {{\\left \\vert}e\\_if\\_i {\\right \\rangle}})(\\sum \\_{i=1}^d \\sqrt{\\lambda \\_i} {{\\left \\langle}e\\_if\\_i {\\right \\vert}})\n+ \\sum \\_{i\\neq j} \\sqrt{\\lambda \\_i\\lambda \\_j}{{\\left \\vert}e\\_if\\_j {\\right \\rangle}}{{\\left \\langle}e\\_if\\_j {\\right \\vert}}.\\end{aligned}$$ \nIn the same way, we can also show the equality ∫02*π*⋯∫02*π**U**θ⃗**Q**U**θ⃗*†*d**θ*1⋯*d**θ**d* = *Q*; *Q* is invariant under the twirling operation.\nFinally, we can calculate $\\int \\_0^{2 \\pi} \\cdots \\int \\_0^{2\n\\pi} U\\_{\\overrightarrow{\\theta}} \\overline{T\\_0} U\\_{\\overrightarrow{\\theta}}^{\\dagger} d \\theta \\_1 \\cdots d \\theta \\_d$ as follows:\n \n$$\\begin{aligned}\n\\fl &\\quad & \\int \\_0^{2 \\pi} \\cdots \\int \\_0^{2\n\\pi} U\\_{\\overrightarrow{\\theta}} \\overline{T\\_0} U\\_{\\overrightarrow{\\theta}}^{\\dagger} d \\theta \\_1 \\cdots d \\theta \\_d \n= \\int \\_0^{2 \\pi} \\cdots \\int \\_0^{2\n\\pi} U\\_{\\overrightarrow{\\theta}} (P+Q) U\\_{\\overrightarrow{\\theta}}^{\\dagger} d \\theta \\_1 \\cdots d \\theta \\_d \\\\\n\\fl &=& \n(\\sum \\_i^d {{\\left \\vert}e\\_if\\_i {\\right \\rangle}}{{\\left \\langle}e\\_if\\_i {\\right \\vert}}) - (\\sum \\_{i=1}^d \\sqrt{\\lambda \\_i} {{\\left \\vert}e\\_if\\_i {\\right \\rangle}})(\\sum \\_{i=1}^d \\sqrt{\\lambda \\_i} {{\\left \\langle}e\\_if\\_i {\\right \\vert}})\n+ \\sum \\_{i\\neq j} \\sqrt{\\lambda \\_i\\lambda \\_j}{{\\left \\vert}e\\_if\\_j {\\right \\rangle}}{{\\left \\langle}e\\_if\\_j {\\right \\vert}} \\\\\n\\fl &\\quad & + \\sum \\_{i\\neq j}\\{ \\sum \\_{k \\neq i,j} \\lambda \\_k + (\\sqrt{\\lambda \\_i} - \\sqrt{\\lambda \\_j})^2\\}{{\\left \\vert}e\\_if\\_j {\\right \\rangle}} {{\\left \\langle}e\\_if\\_j {\\right \\vert}} \\\\\n\\fl &=& (\\sum \\_i^d {{\\left \\vert}e\\_if\\_i {\\right \\rangle}}{{\\left \\langle}e\\_if\\_i {\\right \\vert}}) + (\\sum \\_{i \\neq j} {{\\left \\vert}e\\_if\\_j {\\right \\rangle}}{{\\left \\langle}e\\_if\\_j {\\right \\vert}}) - (\\sum \\_{i=1}^d \\sqrt{\\lambda \\_i} {{\\left \\vert}e\\_if\\_i {\\right \\rangle}})(\\sum \\_{i=1}^d \\sqrt{\\lambda \\_i} {{\\left \\langle}e\\_if\\_i {\\right \\vert}}) \\\\\n\\fl &\\quad & \n- \\sum \\_{i\\neq j} \\sqrt{\\lambda \\_i\\lambda \\_j}{{\\left \\vert}e\\_if\\_j {\\right \\rangle}}{{\\left \\langle}e\\_if\\_j {\\right \\vert}} \\\\\n\\fl &=& I - T,\\end{aligned}$$ \nwhich proves Eq.([average]). ▫\n---\n1. In a strict sense, we can show that there exists an analytical solution for the optimization problem in Eq.([upperbound of two-way t]) by means of Lagrange multiplier. However, even for a 3 × 3 dimensional system, the general solution is too complicated and too ugly to write here.[↩](#fnref1)\n\nTwo-way classical communication remarkably improves local distinguishability\n============================================================================\nWe analyze the difference in the local distinguishability among the following three restrictions; (i) Local operations and only one-way classical communications (one-way LOCC) are permitted. (ii) Local operations and two-way classical communications (two-way LOCC) are permitted. (iii) All separable operations are permitted. We obtain two main results concerning the discrimination between a given bipartite pure state and the completely mixed state with the condition that the given state should be detected perfectly. As the first result, we derive the optimal discrimination protocol for a bipartite pure state in the cases (i) and (iii). As the second result, by constructing a concrete two-way local discrimination protocol, it is proven that the case (ii) is much better than the case (i), i.e., two-way classical communication remarkably improves the local distinguishability in comparison with one-way classical communication at least for a low-dimensional bipartite pure state.\n: state discrimination, one-way LOCC, two-way LOCC, Schmidt rank, Schmidt coefficient\nIntroduction\n============\nRecently, quantum communication has been investigated among many groups as a future technology. Similar to conventional information technology, practical quantum communication technology will require distributed information processing among two or more spatially separated parties. In order to treat this problem, it is necessary to clarify what kind of information processing is possible under respective constraints for permitted operations. In the quantum case, when our quantum system consists of distinct two parties *A* and *B*, we often restrict our operations to local (quantum) operations and classical communications (LOCC) because sending quantum states over long distance is technologically more difficult than sending classical information. Even in this restriction, we can consider the following two formulations; (i) The classical communication is restricted to the direction *A* to *B* (We can similarly treat the restriction of the opposite direction.) (ii) All parties are allowed to communicate classically with each other as much as they like. The case (i) is called the one-way LOCC, and the case (ii) is called the two-way LOCC.\nSince, by definition, the two-way LOCC apparently includes the one-way LOCC, the two-way LOCC is always more powerful than the one-way LOCC in principle. However, due to the following two reasons, it is not easy to characterize the difference between the both performances. As the first reason, it is mathematically hard to rigorously evaluate the performance of a given two-way LOCC protocol, because the mathematical description of two-way LOCC is too complicated. As the second reason, for several simple tasks, the performance of the two-way LOCC was actually shown to be the same as the performance of the one-way LOCC. In fact, there are several settings that has no difference between one-way LOCC and two-way LOCC, e.g., LOCC convertibility of bipartite pure state, Stein’s lemma bound in the simple asymptotic hypothesis testing of the *n*-tensor product of identical states.\nOn the other hand, in several settings, the two-way LOCC is strictly powerful than the one-way LOCC. For example, the distillable entanglement (the amount of maximally entangled states which can be derived from a given state by LOCC) with two-way classical communication is proven to be greater than that with one-way classical communication. Also, this type comparison also has been done by several papers only on the discrimination among orthogonal states. Although several researchers treated this problem, they did not treated the discrimination among non-orthogonal states. In this paper, we compare the both performances quantitatively on the “*local discrimination*” among states that are not necessarily orthogonal, whose purpose is discriminating given states by only LOCC with the *single* copy. In fact, general discrimination problem is closely related to sending classical information via quantum channel and quantum algorithm.\nIn order to quantify the difference of the two-way LOCC and the one-way LOCC, in this paper, we concentrate on a simple setting: the local discrimination of the first state *ρ* on a bipartite system H from the second state *ρ̃* under the condition where the first state *ρ* should be detected perfectly. When the both states *ρ* and *ρ̃* are pure, there is no difference between one-way LOCC and two-way LOCC because any global discrimination protocol can be simulated by one-way LOCC. Surprisingly, as our result, we found that there usually exists non-negligible difference between two restrictions when the second *ρ̃* is the completely mixed state $\\rho\\_{\\rm mix}:=I/\\dim {\\mathcal{H}}$. At the first glance, this setting seems specific, however, due to the following six reasons, it is closely related to several research topics. First, this type analysis produces a bound of the number of perfectly locally distinguishable states. Second, as is explained later, there is a relation between the performance of local distinguishability and amount of entanglement in the case of pure states. Third, this kind of distinguishability is often treated in quantum complexity as Triviality of Coset State."},"data_id":{"kind":"string","value":"arxiv_0185567"}}},{"rowIdx":46397,"cells":{"content":{"kind":"string","value":" 3^{-1} = \\frac{1}{4}, \\\\\n \\frac{{W^{\\*,}}'\\_{t\\_1}(0, {\\phi}\\_{0} )}{\\gamma(W\\_{{\\mathfrak{p}}\\_1})} &= \\frac{2o(t\\_1) + 1}{2} \\log(3) \\text{ when } 2\\mid o(t\\_1)\\end{aligned}$$ \nwhen *μ* ∈ *S*1 as $\\alpha(\\mu, t\\_2) = -\\frac{{\\mathrm{d}}\\_2 {{\\mathrm{Nm}}}(\\mu)}{3\\delta} - t\\_2 \\in \\frac{{\\mathrm{d}}\\_2}{3} - t\\_2 + {\\mathbb{Z}}\\_3 = {\\mathbb{Z}}\\_3$. Since *v*3(Nm(*t*)) = *o*(*t*1*t*2) = *o*(*t*1) − 1, we obtain the lemma when *o*(*t*1) ≥ 0. The case *o*(*t*2) ≥ 0 holds similarly.\nAppendix\n========\nWe record here the set *κ*d, 2(*N*ʹd, 2) ⊂ Ad, 2 = Z/d2Z × Z/2d2Z × Z/2d2Z × Z/d2Z. Note that the group *N*ʹd, 2 and the map d3 ⋅ *κ*d, 2 only depend on d2. This helps with checking Lemma [lemma:additive].\n \n$$\\begin{aligned}\n \\kappa\\_{1, 2}&(N'\\_{1, 2}) =\n \\kappa\\_{3, 2}(N'\\_{3, 2}) =\n\\{\n[0, 0]\n\\},~\n \\kappa\\_{2, 2}(N'\\_{2, 2}) =\n \\kappa\\_{6, 2}(N'\\_{6, 2}) =\n\\{\n [ 1, 0, 0, 1 ],\n[ 1, 2, 2, 1 ]\n\\}.\\end{aligned}$$ \n \n$$\\begin{aligned}\n \\kappa\\_{4, 2}&(N'\\_{4, 2}) =\n \\kappa\\_{12, 2}(N'\\_{12, 2}) =\n \\\\\n\\big\\{\n& [ 1, 2, 6, 3 ],\n[ 1, 6, 2, 3 ],\n[ 1, 0, 0, 1 ],\n[ 1, 4, 4, 1 ],\n[ 3, 6, 2, 1 ],\n[ 3, 2, 6, 1 ],\n[ 3, 0, 0, 3 ],\n[ 3, 4, 4, 3 ]\n\\big\\}.\\end{aligned}$$ \n \n$$\\begin{aligned}\n \\kappa\\_{8, 2}&(N'\\_{8, 2}) = \\\\\\Big\\{\n&[ 1, 2, 14, 7 ],\n[ 1, 6, 10, 7 ],\n[ 1, 10, 6, 7 ],\n[ 1, 14, 2, 7 ],\n[ 1, 0, 0, 1 ],\n[ 1, 4, 12, 1 ],\n[ 1, 8, 8, 1 ],\n[ 1, 12, 4, 1 ],\\\\\n&[ 3, 14, 10, 5 ],\n[ 3, 2, 6, 5 ],\n[ 3, 6, 2, 5 ],\n[ 3, 10, 14, 5 ],\n[ 3, 8, 0, 3 ],\n[ 3, 12, 12, 3 ],\n[ 3, 0, 8, 3 ],\n[ 3, 4, 4, 3 ],\\\\\n&[ 5, 2, 6, 3 ],\n[ 5, 6, 2, 3 ],\n[ 5, 10, 14, 3 ],\n[ 5, 14, 10, 3 ],\n[ 5, 8, 0, 5 ],\n[ 5, 12, 12, 5 ],\n[ 5, 0, 8, 5 ],\n[ 5, 4, 4, 5 ],\\\\\n&[ 7, 14, 2, 1 ],\n[ 7, 2, 14, 1 ],\n[ 7, 6, 10, 1 ],\n[ 7, 10, 6, 1 ],\n[ 7, 0, 0, 7 ],\n[ 7, 4, 12, 7 ],\n[ 7, 8, 8, 7 ],\n[ 7, 12, 4, 7 ]\n\\Big\\}.\\end{aligned}$$ \n \n$$\\begin{aligned}\n \\kappa\\_{24, 2}&(N'\\_{24, 2}) = \\kappa\\_{24, 2}(N'\\_{8, 2}) = 3^{-1} \\cdot \\kappa\\_{8, 2}(N'\\_{8, 2}) =\n\\\\\\Big\\{\n& [ 3, 6, 10, 5 ],\n[ 3, 2, 14, 5 ],\n[ 3, 14, 2, 5 ],\n[ 3, 10, 6, 5 ],\n[ 3, 0, 0, 3 ],\n[ 3, 12, 4, 3 ],\n[ 3, 8, 8, 3 ],\n[ 3, 4, 12, 3 ],\\\\\n&[ 1, 10, 14, 7 ],\n[ 1, 6, 2, 7 ],\n[ 1, 2, 6, 7 ],\n[ 1, 14, 10, 7 ],\n[ 1, 8, 0, 1 ],\n[ 1, 4, 4, 1 ],\n[ 1, 0, 8, 1 ],\n[ 1, 12, 12, 1 ],\\\\\n&[ 7, 6, 2, 1 ],\n[ 7, 2, 6, 1 ],\n[ 7, 14, 10, 1 ],\n[ 7, 10, 14, 1 ],\n[ 7, 8, 0, 7 ],\n[ 7, 4, 4, 7 ],\n[ 7, 0, 8, 7 ],\n[ 7, 12, 12, 7 ],\\\\\n&[ 5, 10, 6, 3 ],\n[ 5, 6, 10, 3 ],\n[ 5, 2, 14, 3 ],\n[ 5, 14, 2, 3 ],\n[ 5, 0, 0, 5 ],\n[ 5, 12, 4, 5 ],\n[ 5, 8, 8, 5 ],\n[ 5, 4, 12, 5 ]\n\\Big\\}.\\end{aligned}$$ \nHere we also include an explicit example for Theorem [thm:intro]. Let *d*1 = − 31, *d*2 = − 127, which have class numbers 3 and 5 respectively and satisfy $d\\_j \\equiv 17 \\bmod{24}$. Then the minimal polynomials of the invariants $f{\\left (}\\left[1, \\tfrac{-1 + \\sqrt{d\\_j}}{2}\\right]{\\right )}$ are\n \n*g*1(*x*) = *x*3 + *x* − 1, *g*2(*x*) = *x*5 − *x*4 − 2*x*3 + *x*2 + 3*x* − 1. \nThe following table lists the values of F(*m*) for various *m*.\n| *a* | *m* | $m \\bmod 96$ | F(*m*) | $\\mathfrak{F}(\\tfrac{m}{2^2})$ | $\\mathfrak{F}(\\tfrac{m}{4^2})$ | $\\mathfrak{F}(\\tfrac{m}{8^2})$ | $\\mathfrak{F}(\\tfrac{m}{16^2})$ | $\\mathfrak{F}(\\tfrac{m}{6^2})$ | $\\mathfrak{F}(\\tfrac{m}{12^2})$ | $\\mathfrak{F}(\\tfrac{m}{24^2})$ | $\\mathfrak{F}(\\tfrac{m}{48^2})$ |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |\n| 1 | 23 ⋅ 3 ⋅ 41 | 24 | 38 | 34 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |\n| 3 | 2 ⋅ 491 | 22 | 4912 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |\n| 5 | 2 ⋅ 3 ⋅ 163 | 18 | 34 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |\n| 7 | 22 ⋅ 35 | 12 | 39 | 33 | 1 | 1 | 1 | 32 | 1 | 1 | 1 |\n| 9 | 22 ⋅ 241 | 4 | 2413 | 241 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |\n| 11 | 2 ⋅ 32 ⋅ 53 | 90 | 532 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |\n| 13 | 2 ⋅ 3 ⋅ 157 | 78 | 34 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |\n| 15 | 25 ⋅ 29 | 64 | 296 | 294 | 292 | 1 | 1 | 1 | 1 | 1 | 1 |\n| 17 | 24 ⋅ 3 ⋅ 19 | 48 | 310 | 36 | 32 | 1 | 1 | 1 | 1 | 1 | 1 |\n| 19 | 2 ⋅ 3 ⋅ 149 | 30 | 34 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |\n| 21 | 2 ⋅ 19 ⋅ 23 | 10 | 234 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |\n| 23 | 22 ⋅ 3 ⋅ 71 | 84 | 36 | 32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |\n| 25 | 22 ⋅ 32 ⋅ 23 | 60 | 233 | 23 | 1 | 1 | 1 | 23 | 1 | 1 | 1 |\n| 27 | 2 ⋅ 401 | 34 | 4012 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |\n| 29 | 2 ⋅ 32 ⋅ 43 | 6 | 432 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |\n| 31 | 23 ⋅ 3 ⋅ 31 | 72 | 38 | 34 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |\n| 33 | 23 ⋅ 89 | 40 | 894 | 892 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |\n| 35 | 2 ⋅ 3 ⋅ 113 | 6 | 34 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |\n| 37 | 2 ⋅ 3 ⋅ 107 | 66 | 34 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |\n| 39 | 22 ⋅ 151 | 28 | 1513 | 151 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |\n| 41 | 22 ⋅ 3 ⋅ 47 | 84 | 36 | 32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |\n| 43 | 2 ⋅ 32 ⋅ 29 | 42 | 292 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |\n| 45 | 2 ⋅ 239 | 94 | 2392 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |\n| 47 | 24 ⋅ 33 | 48 | 310 | 36 | 32 | 1 | 1 | 33 | 3 | 1 | 1 |\n| 49 | 27 ⋅ 3 | 0 | 38 | 36 | 34 | 32 | 1 | 1 | 1 | 1 | 1 |\n| 51 | 2 ⋅ 167 | 46 | 1672 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |\n| 53 | 2 ⋅ 3 ⋅ 47 | 90 | 34 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |\n| 55 | 22 ⋅ 3 ⋅ 19 | 36 | 36 | 32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |\n| 57 | 22 ⋅ 43 | 76 | 433 | 43 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |\n| 59 | 2 ⋅ 3 ⋅ 19 | 18 | 34 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |\n| 61 | 2 ⋅ 33 | 54 | 34 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |\nBy the theorem of Gross-Zagier, one obtains *J*(*d*1, *d*2) by simply takes the product of all the numbers in the fourth column. For *f**s*(*d*1, *d*2), one takes product of the entries $\\mathfrak{F}(\\tfrac{m}{4r^2})$ over all the *m*’s in the table and *r* ∣ *s* satisfying $m \\equiv 4\\cdot 19 (d\\_1 + d\\_2 - 1) \\bmod{4sr}$. This congruence condition eliminates many entries, especially if *s* is large. For example, we have\n \n$$\\begin{aligned}\n f\\_{24}(d\\_1, d\\_2) &= {\\left (}\\mathfrak{F}{\\left (}\\tfrac{2^2 3^5}{4} {\\right )}\\mathfrak{F}{\\left (}\\tfrac{2^4 3^3}{4\\cdot 2^2} {\\right )}\\mathfrak{F}{\\left (}\\tfrac{2^2 3^5}{4 \\cdot 3^2} {\\right )}\\mathfrak{F}{\\left (}\\tfrac{2^4 3^3}{4 \\cdot 6^2} {\\right )}{\\right )}^{1/2} =\n3^4\\end{aligned}$$ \nby Theorem [thm:intro]. One can then immediately check that this is the absolute value of the resultant of the minimal polynomials *g*1, *g*2 in.\n---\n1. The first author is supported by the LOEWE research unit USAG. The second author is partially supported by NSF grant DMS-1762289.[↩](#fnref1)\n2. The Shimura variety is just the product of two modular curves in this case.[↩](#fnref2)\n3. Note that the Fourier expansion of *f* loc. cit. is incorrect.[↩](#fnref3)\n\nOn a Conjecture of Yui and Zagier\n=================================\n[1](#fn1)\nIn this paper, we prove the conjecture of Yui and Zagier concerning the factorization of the resultants of minimal polynomials of Weber class invariants. The novelty of our approach is to systematically express differences of certain Weber functions as products of Borcherds products.\nIntroduction\n============\nIn his book, Weber proved the following well-known theorem in the theory of complex multiplication. For a fundamental discriminant *d* < 0, let $\\mathcal O\\_d={\\mathbb{Z}}[\\theta]$ be the ring of integers of an imaginary quadratic field ${K\\_d}={\\mathbb{Q}}(\\sqrt d)$. Then the CM value of the famous *j*-invariant *j*(*τ*) at *τ* = *θ* is an algebraic integer generating the Hilbert class field of *K**d*. The number *j*(*θ*) is called *singular moduli* and plays an important role in the arithmetic of CM elliptic curves. In the same book, Weber also considered some special modular functions *h* of higher levels and observed that some of their CM values *h*(*θ*) still generate the Hilbert class field of *K**d* (for some choices of *θ*), not the larger class fields as expected for general *h*.\nThese amusing observations were later studied by various authors, see for example,, and. In particular, Gee gave a systematic proof of these facts using Shimura’s reciprocity law. One of them concerns with the CM values of the three classical Weber functions of level 48, which are defined by the following quotients of *η*-functions\n \n$$\\begin{aligned}\n\\label{eq:Weber}\n\\mathfrak f(\\tau) &:=\\zeta\\_{48}^{-1} \\frac{\\eta(\\frac{\\tau+1}2)}{\\eta(\\tau)}=q^{-\\frac{1}{48}} \\prod\\_{n=1}^\\infty (1+q^{n-\\frac{1}2}), \\notag\n\\\\\n\\mathfrak f\\_1(\\tau) &:=\\frac{\\eta(\\frac{\\tau}2)}{\\eta(\\tau)} = q^{-\\frac{1}{48}}\\prod\\_{n=1}^\\infty (1-q^{n-\\frac{1}2}),\n\\\\\n\\mathfrak f\\_2(\\tau) &:= \\sqrt 2 \\frac{\\eta(2\\tau)}{\\eta(\\tau)} = \\sqrt 2 q^{\\frac{1}{24}} \\prod\\_{n=1}^\\infty (1+q^n). \\notag\\end{aligned}$$ \nTogether, they form a 3-dimensional, vector-valued modular function for SL2(Z) (see [eq:Fd]). In fact, the same holds for integral powers of these modular functions (see ). Furthermore, $\\mathfrak f\\_2$ is a modular function for Γ0(2) with character *χ* of order 24:\n \n$$\\label{eq:chi}\n\\mathfrak f\\_2 (\\gamma \\tau) = \\chi(\\gamma) \\mathfrak f\\_2(\\tau), \\quad \\gamma \\in \\Gamma\\_0(2).$$ \nThe kernel of *χ*, denoted by Γ*χ* ⊂ Γ0(2), is a congruence subgroup containing Γ(48) (see ). In, Yui and Zagier studied the CM values of these modular functions. The starting point of their work is the following result.\n[ Proposition] [prop:YZ] Let *d* < 0 be a discriminant satisfying\n \n$$\\label{eqd}\nd \\equiv 1 \\bmod 8, \\hbox{ and } 3\\nmid d.$$ \nDenote *ɛ**d* : = ( − 1)(*d* − 1)/8. For each ideal $\\mathfrak a=[a, \\frac{-b+\\sqrt d}2]$ of the order $\\mathcal O\\_d := {\\mathbb{Z}}[\\tfrac{1 + \\sqrt{d}}{2} ]$ with *a* > 0, let $\\tau\\_{\\mathfrak a} =\\frac{-b+\\sqrt{d}}{2a}$ be the associated CM point and\n \n$$\\label{eq:classinv}\nf(\\mathfrak a) =\\begin{cases}\n \\zeta\\_{48}^{b(a-c-ac^2)} \\mathfrak f(\\tau\\_{\\mathfrak a} ), &{\\operatorname{if }}2|(a, c),\n \\\\\n \\varepsilon\\_d \\zeta\\_{48}^{b(a-c-ac^2) } \\mathfrak f\\_1(\\tau\\_{\\mathfrak a} ), &{\\operatorname{if }}2|a, 2\\nmid c,\n \\\\\n \\varepsilon\\_d \\zeta\\_{48}^{b(a-c+ac^2) } \\mathfrak f\\_2(\\tau\\_{\\mathfrak a} ), &{\\operatorname{if }}2\\nmid a, 2|c.\n \\end{cases}$$ \nThen $f(\\mathfrak a)$ is an algebraic integer depending only on the class of $\\mathfrak a$ in the class group Cl(*d*) of O*d*, i.e. it is a class invariant. Moreover, $H\\_d := {K\\_d}(f(\\mathfrak a)) = {K\\_d}(j(\\tau\\_\\mathfrak{a}))$ is the ring class field of *K**d* corresponding O*d*.\n[rmk:BQF] The class invariant in was defined using binary quadratic forms. It is a standard procedure to go between these and ideals in quadratic fields (see e.g. ).\n[rmk:Gee] The sign *ɛ**d* in the definition of *f*(a) ensures that the class invariants behave nicely under the action of the Galois group. In particular when *d* < 0 is fundamental,\n \n*σ*a2(*f*(a1)) = *f*(a1a2− 1) \nfor any O*d*-ideals a1, a2, where *σ*a ∈ Gal(*H**d*/*K**d*) is associated to the ideal class [a] ∈ Cl(*d*) by Artin’s map. This was conjectured in and proved in.\nThis class invariant is much better than the singular moduli in the sense that its minimal polynomial (class polynomial) has much smaller coefficients. This gives a generator of the Hilbert class field with small height, which is crucial in the speed of elliptic curve primality test. For example, according to, the minimal polynomial of $j(\\frac{1+\\sqrt{-55}}2)$ is \n*x*4 + 335329 ⋅ 134219*x*3 − 375323 ⋅ 101 ⋅ 32987*x*2 + 395711283 ⋅ 101 ⋅ 110641*x* − 31256113293413, \n while the minimal polynomial of $f(\\mathcal O\\_{-55})$ is simply \n*x*4 + *x*3 − 2*x* − 1. \n In, Yui and Zagier made conjectures about the prime factorizations of the discriminants and resultants of such polynomials. The goal of this paper is to prove the conjecture about the factorizations of the resultants, which also clears the path to prove the conjecture about the discriminant (see Remark [rmk:smallCM]).\nFor two co-prime, fundamental discriminants *d*1 and *d*2, Gross and Zagier proved in a beautiful factorization formula for the resultant of the class polynomials of $j(\\frac{d\\_1+\\sqrt{d\\_1}}2)$ and $j(\\frac{d\\_2+\\sqrt{d\\_2}}2)$, which is the norm of the difference $j(\\frac{d\\_1+\\sqrt{d\\_1}}2) -j(\\frac{d\\_2+\\sqrt{d\\_2}}2)$. When $\\left(\\frac{d\\_1d\\_2}p\\right) \\ne -1$, set \n$$\\epsilon(p) =\\begin{cases}\n \\left(\\frac{d\\_1}p\\right) &{\\operatorname{if }}p \\nmid d\\_1,\n \\\\\n \\left(\\frac{d\\_2}p\\right) &{\\operatorname{if }}p \\nmid d\\_2.\n \\end{cases}$$ \n Define in general *ε*(*n*) = ∏*p*∣*n**ε*(*p*)ord*p*(*n*) where ord*p*(*n*) is the power of *p* dividing *n*. For a positive integer *m*, if *ε*(*m*) = − 1, define\n \n$$\\label{eq:Ff}\n \\mathfrak{F}(m) = \\prod\\_{\\substack{n n' = m\\\\ n, n' > 0}} n^{\\epsilon(n')} \\in {\\mathbb{N}},$$ \nwhich is always a prime power. If *ε*(*m*) = 1 or is not defined, define F(*m*) = 1. The result of Gross and Zagier can be stated as follows.\n[Theorem 1.3 in ] [thm:GZ] Let *d*1, *d*2 < 0 be co-prime, fundamental discriminants, and *w**j* = ∣O*d**j*×∣. In the notations above, we have\n \n$$\\label{eq:GZ}\nJ(d\\_1, d\\_2)^2 := \\prod\\_{[\\mathfrak a\\_j] \\in {\\operatorname{Cl}}(d\\_j),~ j = 1, 2} |j(\\tau\\_{\\mathfrak a\\_1}) - j(\\tau\\_{\\mathfrak a\\_2})|^{8/(w\\_1 w\\_2)} =\n\\prod\\_{\\substack{m \\in {\\mathbb{N}},~ a\\in {\\mathbb{Z}}\\\\ a^2 + 4m = d\\_1d\\_2}} \\mathfrak{F}(m).$$ \nInspired by this beautiful formula, Yui and Zagier gave a conjectural formula of the resultant of the minimal polynomials of the Weber class invariants defined above and provided numerical evidences in. This conjecture was originally given using two tables with totally 48 entries (see pg. 1653 of ), but can be simplified and formulated in the following elegant way (see e.g. (14?) in for $d\\_1 \\equiv d\\_2 \\equiv 1 \\bmod{24}$).\n[(14?) in ] [conj:YZ] Let *d*1, *d*2 be co-prime, fundamental discriminants satisfying and *s* ∣ 24. Define the constant\n \n$$\\label{eq:kappa3}\n \\kappa\\_3(s) :=\n \\begin{cases}\n \\tfrac{1}{2},& \\text{ if } (\\tfrac{d\\_1}{3}) = (\\tfrac{d\\_2}{3}) = -1 \\text{ and } 3 \\mid s, \\\\\n1, & \\text{ otherwise,}\n \\end{cases}$$ \nwhich only depends on *d*1, *d*2 and *s*. Then\n \n$$\\label{eq:YZconj}\nf\\_s(d\\_1, d\\_2) := \\prod\\_{[\\mathfrak a\\_j] \\in {\\operatorname{Cl}}(d\\_j),~ j = 1, 2} |f(\\mathfrak a\\_1)^{24/s} - f(\\mathfrak a\\_2)^{24/s}| = \\prod\\_{\\substack{m, a \\in {\\mathbb{N}},~ r \\mid s\\\\ a^2 + 16mr^2 = d\\_1d\\_2\\\\ m \\equiv 19( d\\_1 + d\\_2 - 1) \\bmod{s/r}}} \\mathfrak{F}(m)^{\\kappa\\_3(s)}.$$ \n[rmk:divisibility] Because of the relation *j*(*τ*) = (f224(*τ*) − 16)3/f224(*τ*), we know that *f**s*(*d*1, *d*2) ∣ *J*(*d*1, *d*2) for any co-prime, fundamental discriminants *d*1, *d*2 satisfying. Since the invariants are algebraic integers, it is also clear that *f**s*ʹ(*d*1, *d*2) ∣ *f**s*(*d*1, *d*2) for any *s* ∣ *s*ʹ ∣ 24. The conjecture above also reflects such divisibilities since F(*m*/*r*2) ∣ F(*m*) for all *m*, *r* ∈ N (see e.g. the explicit formula of F(*m*) on pg. 1651 of ).\nWhen *s* = 1, it was suggested in that one can try to prove this conjecture by adapting the analytic approach in with SL2(Z) replaced by Γ0(2). This was later carried out in. More recently, Yang and Yin gave another analytic proof of the conjecture for *s* = 1 in, where the new ingredients are Borcherds’ regularized theta lift in and the big CM formula in. Although the spirits of the approaches are the same, the one in is conceptually easier to understand and opens the door to attack the conjecture for *s* > 1. In this paper, we complete the proof of the conjecture for all *s* ∣ 24.\n[thm:intro] Conjecture [conj:YZ] is true for every *s* ∣ 24.\nFor *s* = 1, the proof of Theorem [thm:intro] in consists of three steps:\n1. Relate $\\mathfrak f\\_2(z\\_1)^{24}- \\mathfrak f\\_2(z\\_2))^{24}$ to a Borcherds product on the Shimura variety associated to the rational quadratic space (*M*2(Q), det). [2](#fn2)\n2. View a pair of CM points (*τ*1, *τ*2) as a big CM point on this Shimura variety in the sense of. Apply the big CM value formula and express the CM value in terms of Fourier coefficients of incoherent Eisenstein series.\n3. Compute the Fourier coefficients in Step (2) and obtain the formula. This is a local calculation.\nIn the first step for *s* = 1, one can find a vector-valued modular function *F̃*1 and identify f2(*z*1)24 − f2(*z*2)24 with the Borcherds product Ψ(*z*1, *z*2, *F̃*1) associated to *F̃*1. Note ${\\mathfrak{f}}\\_2(z)^{24} =2^{12} \\frac{\\Delta(2z)}{\\Delta(z)}$ is a Hauptmodul of Γ0(2), and the Borcherds product Ψ(*z*1, *z*2, *F̃*1) is well-known in the literatures on VOA and moonshine (see e.g. , ). In the second step, one suitably identifies the Galois orbit of CM points with the toric orbit of big CM points, and apply Theorem 5.2 in. This reduces the proof to the third step, where the local calculations have been completed in many special cases (see, and ) and the most general result can be found in Appendix A of.\nTo execute this strategy for *s* > 1, we first need to relate f2(*z*1)24/*s* − f2(*z*2)24/*s* to Borcherds product. Since the function f2(*z*)24/*s* is invariant with respect to Γ*χ*, *s* : = ⟨Γ*χ*, *T**s*⟩ ⊃ Γ(2*s*), one would hope to find the analog of *F̃*1 in *M*!(*ω**s*), with *ω**s* the Weil representation of SL2(Z) on the finite quadratic module associated to the lattice *L**s* (see ), which is the same as the lattice used in to produce Ψ(*z*1, *z*2, *F̃*1), but with the quadratic form scaled by *s*. We have computationally decomposed the representation *ω**s* and analyzed the space of vector-valued modular functions. To our surprise, there is NO modular function whose Borcherds product equals to (f2(*z*1)24/*s* − f2(*z*2)24/*s*)*s*! Our new idea then is to express (f2(*z*1)24/*s* − f2(*z*2)24/*s*)*s* as *a product of Borcherds products*, which works out beautifully.\n[Theorems [theo:BorcherdsLifting] and [thm:Blift]] [theo:factorization] For every d ∣ 24, there is a vector-valued modular function $\\tilde F\\_{\\mathrm{d}}\\in M^!(\\omega\\_{\\mathrm{d}})$ with associated Borcherds product $\\Psi\\_{\\mathrm{d}}(z\\_1, z\\_2) := \\Psi(z\\_1, z\\_2, \\tilde F\\_{\\mathrm{d}})$ such that\n \n$$\\label{eq:product2}\n (\\mathfrak f\\_2(z\\_1)^{24/s} - (\\varepsilon \\mathfrak f\\_2(z\\_2))^{24/s})^s = \\prod\\_{{\\mathrm{d}}\\mid s} \\Psi\\_{\\mathrm{d}}(z\\_1, z\\_2)^{\\varepsilon^{\\frac{24}{\\mathrm{d}}}},$$ \nfor every *s* ∣ 24 and any *ɛ* = ± 1.\nThe index *r* ∣ *s* in the summation on the right hand side of is NOT directly related to the index d ∣ *s* in the product above! Instead, it comes out of local calculation in Section [sec:local].\nEach Borcherds product Ψd(*z*1, *z*2) comes from a different quadratic space depending on d, and is a meromorphic function on the Shimura variety *X*d2, which admits a natural covering map from *X**s*2 when d ∣ *s* (see Section [sect:Borcherds]). One can then pull back Ψd to a function on *X**s*2. Notice that this decomposes the divisor of the LHS, which is a Heegner divisor on *X**s*2, into a sum of pullbacks of Heegner divisors on *X*d2 with d ∣ *s*. When *s* > 1, the product ∏d∣*s*Ψd(*z*1, *z*2) is itself *not* a single Borcherds product on *X**s*2.\nTheorem [theo:factorization] naturally leads one to speculate a generalization of the converse theorem in, i.e. every Heegner divisor on an orthogonal Shimura variety associated to a lattice of signature (*n*, 2) with Witt rank greater than or equal to 2 should be the divisor of a product of Borcherds products.\nTo arrive at this idea, we took *s* = 2 and started from the simple observation that\n \n$$\\label{eq:observation}\n({\\mathfrak{f}}\\_2(z\\_1)^{12} - {\\mathfrak{f}}\\_2(z\\_2)^{12})^2 = ({\\mathfrak{f}}\\_2(z\\_1)^{24} - {\\mathfrak{f}}\\_2(z\\_2)^{24}) \\cdot \\frac{{\\mathfrak{f}}\\_2(z\\_1)^{12} - {\\mathfrak{f}}\\_2(z\\_2)^{12}}{{\\mathfrak{f}}\\_2(z\\_1)^{12} + {\\mathfrak{f}}\\_2(z\\_2)^{12}}.$$ \nWe already know that the first factor on the RHS is a Borcherds product. If we can realize the second factor as a Borcherds product, then the LHS would be a product of Borcherds products (with different quadratic forms). To do that, we can read off the divisor of the second factor, and deduce the principal part of the input to Borcherds’ lift. In this case, it is of the form *q*− 1/2u2 for a suitable vector u2 in a finite dimensional vector space C[A2], where SL2(Z) acts via the Weil representation *ω*2 (see Section [subsec:Lattice] for details). Then we find the irreducible representation in *ω*2 containing u2, which is 3-dimensional, and hope to find the suitable vector-valued modular function *F̃*2 with this principal part. Miraculously, this function exists and its three components are the ( − 24/2)th power of the three Weber functions $\\frac{{\\mathfrak{f}}}{\\sqrt{2}}, \\frac{{\\mathfrak{f}}\\_1}{\\sqrt{2}}, \\frac{{\\mathfrak{f}}\\_2}{\\sqrt{2}}$.\nThe observation generalizes to any *s* ∣ 24 by substituting $X = {\\left (}\\varepsilon \\frac{{\\mathfrak{f}}\\_2(z\\_2)}{{\\mathfrak{f}}\\_2(z\\_1)} {\\right )}^{24/s}$ into the following simple identity in Q(*X*)\n \n(1 − *X*)*s* = ∏d ∣ *s*∏*b* ∣ d(1 − *X**s*/*b*)*b* ⋅ *μ*(d/*b*), \nwhere *μ* is the Möbius function, and multiplying by f2(*z*1)24 on both sides. Note that the identity in holds for any *s* ∈ N (see Lemma [lemma:pol]). Then the miracle continues to happen, and we find a family of vectors {ud : d ∣ 24} (see ) and vector-valued modular functions *F̃*d = *q*− 1/dud + *O*(*q*1/(2d)) producing the Borcherds lifts Ψd (see and Remark [rmk:Fd]). For d > 1, the vector ud satisfies nice invariance properties (see Prop. [prop:udinv]) and is of independent interest, whereas the components of *F̃*d are simply the ( − 24/d)th power of the three Weber functions $\\frac{{\\mathfrak{f}}}{\\sqrt{2}}, \\frac{{\\mathfrak{f}}\\_1}{\\sqrt{2}}, \\frac{{\\mathfrak{f}}\\_2}{\\sqrt{2}}$!\nThe *ɛ* = ± 1 in Theorem [theo:factorization] is there for a good reason. To prove the Yui-Zagier conjecture, we need to choose $\\varepsilon = \\varepsilon\\_{d\\_1}\\varepsilon\\_{d\\_2} = (-1)^{\\frac{d\\_1 +d\\_2 -2}8}$ (see Proposition [prop:CMFormula] and its proof). It is also amusing to see that the same *ɛ* appears when we calculate the Fourier coefficients of derivatives of certain Einsenstein series (see Theorem [thm:main1]).\nTo complete the proof, we can now apply the second step to each Borcherds product, obtain a big CM value formula, and add them together. Note that the identification of the Galois orbit of (*τ*a1, *τ*a2) used in defining *f**s*(*d*1, *d*2) with the big CM cycle in depends on the input *F̃*d in step one. Therefore, it is not a priori clear that this will work out. We prove this in Proposition [prop:CMFormula], which crucially depends on Lemma [lemma:compact]. This unexpected result was first observed with some computer calculations, and has been reduced to a computation with finite groups in GL2(Z/3Z) and GL2(Z/16Z). Finally, we apply the local calculations in to finish off Step (3).\n[rmk:smallCM] With Theorem [theo:factorization], one can now replace the big CM value formula in with the small CM value formula in to prove the conjectural factorization of the discriminant of the minimal polynomials of the Weber invariants in. In fact, Yui and Zagier wrote the discriminant as the norm of an algebraic integer in the Hilbert class field and gave a conjectural prime factorization of the ideal generate by it. This refined conjecture can be studied by analyzing the arithmetic intersection calculations in, which are generalizations of the algebraic approach in. Theorem [theo:factorization] tells us how to decompose the divisor of modular function (f2(*z*1)24/*s* − (*ɛ*f2(*z*2))24/*s*)*s* into sums of Heegner divisors, which will be vital in attacking this refined conjecture in. We plan to carry these out as a sequel to this work.\nThis paper is organized as follows. After setting up notation and defining basic terms in Section [sect2], we study in Section [sect3] the action of certain subgroup *H*dʹ ⊂ SO(*L*d)/Γ*L*d on the finite quadratic module Ad : = *L*d∨/*L*d and use it to decompose the Weil representation *ω*d of SL2(Z) on C[Ad]*H*dʹ. The goal and main result is to construct certain element ${\\mathfrak{u}}\\_{\\mathrm{d}}\\in {\\mathbb{C}}[\\mathcal A\\_{\\mathrm{d}}]^{H\\_{\\mathrm{d}}'}$ satisfying ([eq:ufdinv]). This vector generates a 3-dimensional, *H*ʹd-invariant subrepresentation of *ω*d, and will be crucial in finding the input *F̃*d that produces the Borcherds product Ψd. In Section [sect:Borcherds], we view product of two modular curves as a Shimura variety of orthogonal type (2, 2) associated to *L*d, construct the Borcherds product Ψd, and prove Theorem [theo:factorization]. In Section [sect:BigCM], we view the pair $(\\tau\\_{\\mathfrak a\\_1}, \\tau\\_{\\mathfrak a\\_2})$ as a big CM point on the product of two modular curves and study its Galois orbit. The upshot is Prop. [prop:CMFormula], which relates the left hand side of Conjecture [conj:YZ] to the big CM value of Borcherds products. By the second step of strategy, Conjecture [conj:YZ] is reduced to local calculation of certain Eisenstein series and its derivative, which we carry out in Section [subsec:local] using the results in the appendix of. Finally in the appendix, we explicitly write down the cosets in the finite quadratic module used in constructing the Borcherds products, and include a numerical example for *d*1 = − 31 and *d*2 = − 127.\n*Acknowledgement*: We thank Hongbo Yin for a careful reading and helpful comments of an earlier version of this paper.\nPreliminaries\n=============\nWeil Representation.\n--------------------\nLet (*L*, *Q*) be an even integral lattice of signature (2, 2) and *V* : = *L* ⊗ Q the rational quadratic space. Denote *L*ʹ the dual lattice and A*L* : = *L*ʹ/*L* the finite quadratic module. The group SL2(Z) acts on *U**L* : = C[A*L*] via the Weil representation *ω**L* given by\n \n$$\\label{eq:omegaL}\n \\omega\\_L(T){\\mathfrak{e}}\\_h = {{\\mathbf{e}}}(-Q(h)) {\\mathfrak{e}}\\_h,~ \\omega\\_L(S) {\\mathfrak{e}}\\_h = \\frac{1}{\\sqrt{|{{\\mathcal{A}}}\\_L|}} \\sum\\_{\\mu \\in {{\\mathcal{A}}}\\_L} {{\\mathbf{e}}}((\\mu, h)) {\\mathfrak{e}}\\_\\mu,$$ \nwhere {e*μ* : *μ* ∈ A*L*} is the standard basis of *U**L* and\n \n$$\\label{eq:TS}\n T:= {\\begin{pmatrix}\n 1 & 1\\\\\n 0 & 1\n\\end{pmatrix}},\\quad S:= {\\begin{pmatrix}\n 0 & -1\\\\\n 1 & 0\n\\end{pmatrix}}.$$ \nNote that this differs from the convention of Borcherds by complex conjugation.\nLet *S*(*L*) = ⊕ *μ* ∈ *L*ʹ/*L**ϕ**μ* ⊂ *S*(*V* ⊗ A*f*) with $\\hat{L} = L \\otimes \\hat{{\\mathbb{Z}}}$ and \n$$\\phi\\_\\mu={\\operatorname{Char}}(\\mu+\\hat L).$$ \n Under the isomorphism *U**L* → *S*(*L*) that maps e*μ* to *ϕ**μ*, the representation *ω**L* becomes the restriction of the Weil representation *ω* = *ω**V*, *ψ* (with the usual idelic character *ψ* of Q) from SL2(A) to (the diagonally embedded) SL2(Z). We will sometimes switch the representation spaces between *U**L* and *S*(*L*).\nWeber Functions.\n----------------\nFor any finite dimensional, C-representation *ρ* : Γ → *V* of a finite index subgroup Γ ⊂ SL2(Z), denote *M*!(*ρ*, Γ) the space of weakly holomorphic, vector-valued modular function with respect to *ρ*. We drop *ρ*, resp. Γ, from the notation if *ρ* is trivial, resp. Γ = SL2(Z). For example, the three Weber functions defined by ([eq:Weber]) form a vector-valued modular function $\\begin{pmatrix}\n {\\mathfrak{f}}\\_2 \\\\ {\\mathfrak{f}}\\_1 \\\\ {\\mathfrak{f}}\\end{pmatrix} \\in M^!(\\overline{\\varrho\\_{24}})$. Here, for a positive integer d and *j* ∈ (Z/2dZ)×, the representation $\\varrho\\_{{\\mathrm{d}}, j}:{\\operatorname{SL}}\\_2({\\mathbb{Z}}) \\to {{\\mathrm{GL}}}\\_3({\\mathbb{C}})$ is defined by\n \n$$\\label{eq:varrhod}\n \\varrho\\_{{\\mathrm{d}}, j}(T) :=\n \\begin{pmatrix}\n \\zeta\\_{{\\mathrm{d}}}^{-j} & 0 & 0 \\\\\n0 & 0 & \\zeta\\_{2{\\mathrm{d}}}^{j}\\\\\n0 & \\zeta\\_{2{\\mathrm{d}}}^{j} & 0\n \\end{pmatrix},~\n \\varrho\\_{{\\mathrm{d}}, j}(S) :=\n \\begin{pmatrix}\n0 & 1 & 0 \\\\\n1 & 0 & 0 \\\\\n0 & 0 & 1\n \\end{pmatrix}.$$ \nWe simply write $\\varrho\\_{\\mathrm{d}}$ for $\\varrho\\_{{\\mathrm{d}}, 1}$. Finally, $\\bar{\\rho}(g)= \\overline{\\rho(g)}$. Later, the modular function\n \n$$\\label{eq:Fd}\n F\\_{\\mathrm{d}}(\\tau) :=\n \\sqrt{2}^{24/{\\mathrm{d}}}\n\\begin{pmatrix}\n {\\mathfrak{f}}\\_2^{-24/{\\mathrm{d}}}(\\tau) \\\\ {\\mathfrak{f}}\\_1^{-24/{\\mathrm{d}}}(\\tau) \\\\ {\\mathfrak{f}}^{-24/{\\mathrm{d}}}(\\tau)\n\\end{pmatrix}\n\\in M^!(\\varrho\\_{\\mathrm{d}}),~ {\\mathrm{d}}\\mid 24$$ \nwill play an important role for us as the representation $\\varrho\\_{\\mathrm{d}}$ defined above is a subrepresentation of certain Weil representation that we will consider.\n[rmk:FC] For convenience later, we will denote\n \n$$\\label{eq:adn}\n \\sqrt{2}^{24/{\\mathrm{d}}} {\\mathfrak{f}}\\_2^{-24/{\\mathrm{d}}}(\\tau) = {\\left (}\\frac{\\eta(\\tau)}{\\eta(2\\tau)}{\\right )}^{24/{\\mathrm{d}}} = \\sum\\_{l \\ge -1,~ l \\equiv -1 \\bmod{{\\mathrm{d}}}} c\\_{\\mathrm{d}}(l) q^{l/{\\mathrm{d}}}\\in q^{-1/{\\mathrm{d}}}{\\mathbb{Z}}\\llbracket q \\rrbracket.$$ \nClearly *c*d( − 1) = 1 for every d ∣ 24. We will also denote *c*− 1(*l*) the *l*th Fourier coefficient of $2^{12}{\\mathfrak{f}}\\_1^{-24}(2\\tau) = {\\mathfrak{f}}\\_2^{24}(\\tau) = 2^{12} \\frac{\\Delta(2\\tau)}{\\Delta(\\tau)}$.\nLet *χ* : Γ0(2) → C× be the character defined in. On the generators *T*, *S*2 and *T**B* of Γ0(2), where\n \n$$\\label{eq:B}\nB:=ST^2S^{-1} = {\\begin{pmatrix}\n 1 & 0\\\\\n -2 & 1\n\\end{pmatrix}},$$ \nthe character *χ* is explicitly given by\n \n$$\\label{eq:chi0}\n \\chi(T) = \\zeta\\_{24}, ~ \\chi(S^2)=1, \\, \\hbox{ and }~ \\chi(TB) = 1.$$ \nThe kernel of *χ* is a normal subgroup of Γ0(2) defined by\n \nΓ*χ* : = ⟨Γ0(2)der, *T*24, *S*2, *T**B*⟩ ⊂ Γ0(2), \nwhere Γ0(2)der is the derived subgroup of Γ0(2). We remark that Γ*χ* is the group Φ00(24) in. Furthermore, it contains the congruence subgroup Γ0(48) ∩ Γ(24) and Γ0(2)/Γ*χ* ≅ Z/24. More generally, for any divisor d ∣ 24, denote the kernel of *χ*24/d by\n \nΓ*χ*, d : = ⟨Γ*χ*, *T*d⟩ ⊂ Γ0(2). \nIt has index d in Γ0(2) and contains Γ*χ* = Γ*χ*, 24, as well as the congruence subgroup\n \nΓd : = Γ1(2d) ∩ Γ(d). \nIn particular, Γ0(2) = Γ*χ*, 1. More generally for d ∣ dʹ ∣ 24, we have Γd ⊃ Γdʹ. For future convenience, we also write d*p* for the *p*-part of d. Then clearly d = d2d3.\nDecomposition of Weil Representations\n=====================================\nLattice\n-------\nFor a divisor d ∣ 24, consider the quadratic lattice\n \n$$\\label{eq:Ld}\nL\\_{\\mathrm{d}}= \\left\\{\n\\lambda = {\\begin{pmatrix}\n \\lambda\\_{00} & \\lambda\\_{01}\\\\\n 2\\lambda\\_{10} & \\lambda\\_{11}\n\\end{pmatrix}}: \\lambda\\_{ij} \\in {\\mathbb{Z}}\\right\\},~ Q\\_d(\\lambda) := {\\mathrm{d}}\\det(\\lambda).$$ \nThe dual lattice is given by\n \n$$\\label{eq:Ld'}\nL\\_{\\mathrm{d}}' = \\left\\{ \\lambda = \\frac{1}{{\\mathrm{d}}} {\\begin{pmatrix}\n \\lambda\\_{00} & \\lambda\\_{01}/2\\\\\n \\lambda\\_{10} & \\lambda\\_{11}\n\\end{pmatrix}}: \\lambda\\_{ij} \\in {\\mathbb{Z}}\\right\\}.$$ \nThe finite quadratic module *L*dʹ/*L*d is then isomorphic to\n \nAd : = {*h* = [*h*0, *h*1, *h*2, *h*3] : *h*0, *h*3 ∈ Z/dZ, *h*1, *h*2 ∈ Z/(2dZ)}, \nwhere the isomorphism is fixed throughout and given by\n \n$$\\label{eq:Acdiso}\n \\begin{split}\n L\\_{\\mathrm{d}}' /L\\_{\\mathrm{d}}&\\cong {{\\mathcal{A}}}\\_{\\mathrm{d}}\\\\\n \\frac{1}{{\\mathrm{d}}} {\\begin{pmatrix}\n \\lambda\\_{00} & \\lambda\\_{01}/2\\\\\n \\lambda\\_{10} & \\lambda\\_{11}\n\\end{pmatrix}} + L\\_{\\mathrm{d}}&\\mapsto \\left[ \\lambda\\_{00}, \\lambda\\_{01}, \\lambda\\_{10}, \\lambda\\_{11}\\right].\n \\end{split}$$ \nVia this isomorphism, the quadratic form *Q*d on Ad becomes\n \n$$\\label{eq:Qd}\n Q\\_{\\mathrm{d}}(h) := \\frac{2h\\_0 h\\_3 - h\\_1h\\_2}{2{\\mathrm{d}}} \\in \\frac{1}{2{\\mathrm{d}}}{\\mathbb{Z}}/{\\mathbb{Z}}$$ \nfor *h* = [*h*0, *h*1, *h*2, *h*3] ∈ Ad. We can now map *L*d into *L*dʹ/*L*d ≅ Ad via\n \n$$\\label{eq:kappad}\n \\begin{split}\n \\kappa\\_{\\mathrm{d}}: L\\_{\\mathrm{d}}&\\to L\\_{\\mathrm{d}}'/L\\_{\\mathrm{d}}\\\\\n \\lambda &\\mapsto \\frac{1}{{\\mathrm{d}}} \\lambda + L\\_{\\mathrm{d}}\\end{split}$$ \nwhich is compatible with the left and right action of Γ0(2), i.e.\n \n*g*1 ⋅ *κ*d(*λ* ⋅ *g*2) = *κ*d(*g*1 ⋅ *λ* ⋅ *g*2) = *κ*d(*g*1 ⋅ *λ*) ⋅ *g*2 \nfor all *g*1, *g*2 ∈ Γ0(2) and *λ* ∈ *L*d. By viewing Γ*χ*, 1 = Γ0(2) as a subset of *L*d, we can send it to a subset in Ad. If we denote\n \nAd0 : = {[*h*0, *h*1, *h*2, *h*3] ∈ Ad : *h*2 = 0 ∈ Z/(2dZ)}, \nit will be helpful to know the parts of Γ0(2) that land in Ad0 under *κ*d when we simplify the expression of Borcherds products. For this we need the following lemma, whose proof will follow from combining the corresponding local results in Lemma [lemma:j3] and [lemma:j2].\n[lemma:j] For any *j* ∈ Z/dZ, we have (viewing Γ0(2) ⊂ *L*d) \n*κ*d(*T**j*Γ*χ*, d) ∩ Ad0 = {d3− 1[*r*, *r*(2*j* + (*r*2 − 1)), 0, *r*] : *r* ∈ (Z/dZ)×}. \nNote that $r^3 - r \\bmod {2{\\mathrm{d}}}$ is well-defined for *r* ∈ (Z/dZ)× when d ∣ 24. Furthermore \n$$r^3 - r \\equiv\n \\begin{cases}\n {\\mathrm{d}}\\bmod{2{\\mathrm{d}}},& \\text{if } 8 \\mid {\\mathrm{d}}\\text{ and } r \\equiv \\pm 3 \\bmod{8}, \\\\\n 0 \\bmod{2{\\mathrm{d}}},& \\text{otherwise.}\n \\end{cases}$$ \nWe want to analyze the action of the finite orthogonal group *H*d : = SO(*L*d)/Γ*L*d on Ad, which intertwines the Weil action of (*G* = SL2(Z), *ω*d = *ω**L*d) on\n \n*U*d : = C[Ad], \nand will be useful for us to decompose the Weil representation *ω*d. In particular, we want to consider a subrepresentation of *ω*d on the subspace *U*d*H*ʹd ⊂ *U*d fixed by a subgroup *H*ʹd ⊂ *H*d.\nLet GL2(Q) × GL2(Q) acts on *V*d = *L*d ⊗ Q = *M*2(Q) via \n(*g*1, *g*2) ⋅ *X* = *g*1*X**g*2− 1. \n This action gives an identification of GSpin(*V*) with *H* = {(*g*1, *g*2) : det*g*1 = det*g*2}, and a commutative diagram of exact sequences: \n$$\\xymatrix{\n 1 \\ar[r] &\\{\\pm 1\\} \\ar[r] \\ar[d] &{\\operatorname{SL}}\\_2 \\times {\\operatorname{SL}}\\_2 \\ar[r] \\ar[d] &{{\\mathrm{SO}}}(V) \\ar[r] \\ar[d] &1\n\\cr\n 1 \\ar[r] &{\\mathbb{G}}\\_m \\ar[r] &H \\ar[r] &{{\\mathrm{SO}}}(V) \\ar[r] &1.\n \\cr\n}$$ \nFor the particular lattice *L*d, we have\n \n$$\\label{eq:disck}\n {{\\mathrm{SO}}}(L\\_{\\mathrm{d}}) = \\overline{\\Gamma\\_0(2) \\times \\Gamma\\_0(2)}= \\Gamma\\_0(2) \\times \\Gamma\\_0(2)/\\{ \\pm (I\\_2, I\\_2)\\},~$$ \nAs { ± (*I*2, *I*2)} does not matter in this paper, we will simply identify SO(*L*d) with Γ0(2) × Γ0(2) and drop the over line. Under this identification, we have Γ*L*d = Γd × Γd, where Γd is defined in. In particular, we are interested in the action of the subgroup of SO(*L*d) generated by the images of (*T*, *T*) and Γ*χ*, d × Γ*χ*, d. We let *H*dʹ be its image in *H*d = *N*d × *N*d, where *N*d : = Γ0(2)/Γd. Let *N*ʹd : = Γ*χ*, d/Γd, then we have\n \n*N*dʹ × *N*dʹ ⊂ *H*ʹd ⊂ *H*d = *N*d × *N*d. \nSince Γ0(2)/Γ*χ*, d ≅ Z/dZ, the quotient group *H*d/*H*ʹd is isomorphic to Z/dZ. For prime *p* and an abelian group *K*, we denote *K**p* : = *K* ⊗ ZZ*p*. By carrying this through product and quotient, we can also make sense of *K**p* for *K* ∈ {*N*d, *N*dʹ, *H*d, *H*dʹ}. Since d is only divisible by 2 and 3 in our case, the Chinese remainder theorem implies\n \n$$\\begin{aligned}\nH\\_{\\mathrm{d}}&\\cong H\\_{{\\mathrm{d}}, 2} \\times H\\_{{\\mathrm{d}}, 3},~ H'\\_{\\mathrm{d}}\\cong H'\\_{{\\mathrm{d}}, 2} \\times H'\\_{{\\mathrm{d}}, 3},\n\\\\\n {N'\\_{{\\mathrm{d}}, p} \\times N'\\_{{\\mathrm{d}}, p}}&\\subset H'\\_{{\\mathrm{d}}, p}\\subset H\\_{{\\mathrm{d}}, p} = {N\\_{{\\mathrm{d}}, p} \\times N\\_{{\\mathrm{d}}, p}}.\\end{aligned}$$ \nFor the same reason, we have the decomposition\n \nAd ≅ Ad, 2 × Ad, 3, Ad, *p* : = Ad ⊗ ZZ*p*. \nUsing this isomorphism, we can write *ω*d ≅ *ω*d, 2 ⊗ *ω*d, 3 and *U*d ≅ *U*d, 2 ⊗ *U*d, 3, where *ω*d, *p* is the Weil representation of SL2(Z*p*) acting on *U*d, *p* associated to Ad, *p*.\nNow, we introduce the vector ud ∈ *U*d.\n \n$$\\label{eq:ufd}\n \\begin{split}\n {\\mathfrak{u}}\\_{\\mathrm{d}}&:= {\\mathfrak{u}}\\_{{\\mathrm{d}}, 2} \\otimes {\\mathfrak{u}}\\_{{\\mathrm{d}}, 3} = \\sum\\_{j \\in {\\mathbb{Z}}/{\\mathrm{d}}{\\mathbb{Z}}} a\\_{\\mathrm{d}}(j) {\\left (}\\sum\\_{h \\in \\kappa\\_{{\\mathrm{d}}}(T^j \\Gamma\\_{\\chi, {\\mathrm{d}}})} {\\mathfrak{e}}\\_h {\\right )},\\\\\na\\_{\\mathrm{d}}(j) &:= {\\left (}\\sum\\_{s \\in ({\\mathbb{Z}}/{\\mathrm{d}}{\\mathbb{Z}})^\\times} \\zeta\\_{{\\mathrm{d}}}^{sj} {\\right )}= \\mu{\\left (}\\frac{{\\mathrm{d}}}{({\\mathrm{d}}, j)} {\\right )}\\frac{\\varphi({\\mathrm{d}})}{\\varphi({\\mathrm{d}}/({\\mathrm{d}}, j))} \\in {\\mathbb{Z}},\n \\end{split}$$ \nwhere ud, *p* = ud, *p*(1, …, 1) ∈ *U*d, *p*ʹ ⊂ *U*d, *p* is the vector defined in and, *μ* and *φ* are the Möbius and Euler *φ*-function respectively. Note that *a*d(*j*) is defined for any d ∈ N and *j* ∈ Z/dZ.\nA natural question is where the element ud, *p* comes from and what it is good for? In the next two subsections, we will give some ideas where they come from. For now, we are satisfied to give its nice properties as below. See Prop. [prop:repembed] below.\n[lemma:ufinv] For any d, *r* ∣ 24, the vector ud ∈ Z[Ad] is invariant with respect to Γ*χ*, *r* × Γ*χ*, *r* if and only if d ∣ *r*.\nIf d ∣ *r*, then Γ*χ*, *r* ⊂ Γ*χ*, d and we just need to prove the case when *r* = d. Let (*g*1, *g*2) ∈ SO(*L*d) with *g**j* ∈ Γ*χ*, d. Then\n \n$$\\begin{aligned}\n{ (g\\_1, g\\_2)} \\cdot {\\mathfrak{u}}\\_{\\mathrm{d}}&= \\sum\\_{j \\in {\\mathbb{Z}}/{\\mathrm{d}}{\\mathbb{Z}}} a\\_{\\mathrm{d}}(j) {\\left (}\\sum\\_{h \\in \\kappa\\_{{\\mathrm{d}}}(g\\_1 T^j \\Gamma\\_{\\chi, {\\mathrm{d}}} g\\_2^{-1})} {\\mathfrak{e}}\\_h {\\right )}= {\\mathfrak{u}}\\_{\\mathrm{d}},\n \\end{aligned}$$ \nwhere we have used the fact that Γ*χ*, d is normal in Γ0(2) with coset representatives {*T**j* : *j* ∈ Z/dZ}. Similarly, (*T*, *T*) ⋅ ud = ud. Therefore ud is Γ*χ*, d × Γ*χ*, d-invariant. If ${\\mathrm{d}}\\nmid r$, then (*T**r*, 1) ∈ (Γ*χ*, *r* × Γ*χ*, *r*) ∖ (Γ*χ*, d × Γ*χ*, d). It is easy to see that \n(*T**r*, 1) ⋅ ud = ∑*j* ∈ Z/dZ*a*d(*j* − *r*)(∑*h* ∈ *κ*d(*T**j**N*ʹd)e*h*) ≠ u*d* \n since $\\frac{a\\_{\\mathrm{d}}(-r)}{\\varphi({\\mathrm{d}})} = \\frac{\\mu({\\mathrm{d}}/({\\mathrm{d}}, r))}{\\varphi({\\mathrm{d}}/({\\mathrm{d}}, r))} \\neq 1 = \\frac{a\\_{\\mathrm{d}}(0)}{\\varphi({\\mathrm{d}})}$ when ${\\mathrm{d}}\\nmid r$.\n[prop:udinv] For any d ∣ 24, we have\n \n*ω*d(*g*)ud = *χ*(*g*)− 24/dud \nfor all *g* ∈ Γ0(2).\nThis follows directly from the local results [prop:udinv3] and [prop:decomp2] as \n*ω*d(*g*)ud = (*ω*d, 2(*g*)ud, 2) ⊗ (*ω*d, 3(*g*)ud, 3) = *χ*(*g*)− 24(d2/d3 + d3/d2)ud, 2 ⊗ ud, 3 = *χ*(*g*)− 24/dud. \n for all *g* ∈ Γ0(2). Here we have used $\\frac{1}{{\\mathrm{d}}} - (\\frac{{\\mathrm{d}}\\_2}{{\\mathrm{d}}\\_3} + \\frac{{\\mathrm{d}}\\_3}{{\\mathrm{d}}\\_2}) \\in {\\mathbb{Z}}$ when d ∣ 24.\nNow, define two further vectors\n \nvd : = *ω*d(*S*)ud, wd : = *ζ*2d− 1*ω*d(*T*)vd. \nNote that ud, vd and wd are linearly independent for all d ∣ 24. The key to the input of Borcherds lifting is then constructed using these vectors in the following result.\n[prop:repembed] The representations $\\varrho\\_{\\mathrm{d}}$ defined in is a subrepresentation of the Weil representation *ω*d via the map\n \n$$\\label{eq:iotad}\n \\begin{split}\n \\iota\\_{\\mathrm{d}}: {\\mathbb{C}}^3 &\\to U\\_{\\mathrm{d}}^{H\\_{\\mathrm{d}}'} \\subset U\\_{\\mathrm{d}}\\\\\n \\begin{pmatrix}\n a \\\\ b \\\\ c\n\\end{pmatrix} &\\mapsto a{\\mathfrak{u}}\\_{\\mathrm{d}}+ b {\\mathfrak{v}}\\_{\\mathrm{d}}+ c{\\mathfrak{w}}\\_{\\mathrm{d}}.\n \\end{split}$$ \nLet *F*d be the modular function defined in. The function *ι*d ∘ *F*d is then in *M*!(*ω*d) and invariant with respect to the orthogonal group *H*dʹ ⊂ SO(*L*d)/Γ*L*d. Furthermore, it has the principal part\n \n$$\\label{eq:Fdprin}\n \\iota\\_d \\circ F\\_{\\mathrm{d}}(\\tau) = q^{-1/{\\mathrm{d}}} {\\mathfrak{u}}\\_{\\mathrm{d}}+\n \\begin{cases}\n O(q^{1/2}),& {\\mathrm{d}}> 1,\\\\\n O(1),& {\\mathrm{d}}= 1,\\\\\n \\end{cases}$$ \n[rmk:Fd] When d = 1, the function *ι*d ∘ *F*d differs from the input in by a constant vector. To simplify the notation, we will write\n \n$$\\label{eq:Fdt}\n \\tilde{F}\\_{\\mathrm{d}}:= \\iota\\_{\\mathrm{d}}\\circ F\\_{\\mathrm{d}}+\n \\begin{cases}\n 24({\\mathfrak{e}}\\_{(0, 0)} + {\\mathfrak{e}}\\_{(1/2, 0)}),& {\\mathrm{d}}= 1,\\\\\n 0, & {\\mathrm{d}}> 1,\n \\end{cases}$$ \nwhich is an element in *M*!(*ω*d) invariant with respect to *H*dʹ.\nIt suffices to check on the generators *T*, *S* of SL2(Z). From the definition and Proposition [prop:udinv], it is clear that\n \n$$\\begin{aligned}\n \\omega\\_{\\mathrm{d}}(T) {\\mathfrak{u}}\\_{\\mathrm{d}}&= \\zeta^{-1}\\_{{\\mathrm{d}}} {\\mathfrak{u}}\\_{\\mathrm{d}},~\n\\omega\\_{\\mathrm{d}}(T) {\\mathfrak{v}}\\_{\\mathrm{d}}= \\zeta\\_{2{\\mathrm{d}}} {\\mathfrak{w}}\\_{\\mathrm{d}},\\\\\n\\omega\\_{\\mathrm{d}}(T) {\\mathfrak{w}}\\_{\\mathrm{d}}&= \\zeta^{-1}\\_{2{\\mathrm{d}}} \\omega\\_{\\mathrm{d}}(T^2S){\\mathfrak{u}}\\_{\\mathrm{d}}=\n\\zeta^{-1}\\_{2{\\mathrm{d}}} \\omega\\_{\\mathrm{d}}(SB){\\mathfrak{u}}\\_{\\mathrm{d}}= \\zeta\\_{2{\\mathrm{d}}} \\omega\\_{\\mathrm{d}}(S){\\mathfrak{u}}\\_{\\mathrm{d}}= \\zeta\\_{2{\\mathrm{d}}} {\\mathfrak{v}}\\_{\\mathrm{d}},\\\\\n\\omega\\_{\\mathrm{d}}(S){\\mathfrak{u}}\\_{\\mathrm{d}}&= {\\mathfrak{v}}\\_{\\mathrm{d}},~\n\\omega\\_{\\mathrm{d}}(S){\\mathfrak{v}}\\_{\\mathrm{d}}= {\\mathfrak{u}}\\_{\\mathrm{d}},\\\\\n\\omega\\_{\\mathrm{d}}(S){\\mathfrak{w}}\\_{\\mathrm{d}}&= \\zeta\\_{2{\\mathrm{d}}}^{-1} \\omega\\_{\\mathrm{d}}(STS){\\mathfrak{u}}\\_{\\mathrm{d}}=\n\\zeta\\_{2{\\mathrm{d}}}^{-1} \\omega\\_{\\mathrm{d}}(STSTBS^2){\\mathfrak{u}}\\_{\\mathrm{d}}= \\zeta\\_{2{\\mathrm{d}}}^{-1} \\omega\\_{\\mathrm{d}}(STSTSTS){\\mathfrak{u}}\\_{\\mathrm{d}}= \\zeta\\_{2{\\mathrm{d}}}^{-1} \\omega\\_{\\mathrm{d}}(S){\\mathfrak{u}}\\_{\\mathrm{d}}= {\\mathfrak{w}}\\_{\\mathrm{d}}.\\end{aligned}$$ \nIn the following two subsections, we work at the 2-part and 3-part separately and construct ud, *p* for *p* = 2, 3. This will shed some light on where ud comes from.\nThe case *p* = 3.\n-----------------\nThere are two possibilities for Ad, 3. If $3 \\nmid {\\mathrm{d}}$, then Ad, 3 is trivial. If 3 ∣ d, we can identify the groups Ad, 3 and A : = *M*2(F3) via\n \n$$\\label{eq:Ac3}\n \\begin{split}\n \\kappa\\_{{\\mathrm{d}}, 3}:M\\_2({\\mathbb{F}}\\_3) &\\cong {{\\mathcal{A}}}\\_{{\\mathrm{d}}, 3} \\\\\n {\\begin{pmatrix}\n h\\_0 & -h\\_1\\\\\n h\\_2 & h\\_3\n\\end{pmatrix}} \\bmod{3} &\\mapsto h =[{h\\_0},{h\\_1},{h\\_2},{h\\_3}] \\otimes {\\mathbb{Z}}\\_3,\n \\end{split}$$ \nwhich is just the map *κ*d in tensored with Z3. This is an isomorphism of finite quadratic modules if we equip *M*2(F3) with the quadratic form *Q*d, 3 : = (3d2)− 1det, which has value in $\\frac{1}{3}{\\mathbb{Z}}/{\\mathbb{Z}}$. Then *H*3 ≅ SL2(F3) × SL2(F3), *H*ʹ3 = ⟨*N*ʹ3 × *N*ʹ3, (*T*, *T*)⟩, where\n \n$$\\label{eq:N3'}\n \\begin{split}\nN\\_{{\\mathrm{d}}, 3}' &:=\n\\left\\{\n\\begin{smat}\n{1}{0}{0}{1}\n\\end{smat},\n\\begin{smat}\n { -1 }{ -1 }{ -1 }{ 1 }\n\\end{smat},\n\\begin{smat}\n { 0 }{ 1 }{ -1 }{ 0 }\n\\end{smat},\n\\begin{smat}\n { -1 }{ 1 }{ 1 }{ 1 }\n\\end{smat},\n\\begin{smat}\n { -1 }{ 0 }{ 0 }{ -1 }\n\\end{smat},\n\\begin{smat}\n { 1 }{ 1 }{ 1 }{ -1 }\n\\end{smat},\n\\begin{smat}\n { 0 }{ -1 }{ 1 }{ 0 }\n\\end{smat},\n{\\left(\\begin{smallmatrix}\n 1 & -1 \\\\\n -1 & -1 \n\\end{smallmatrix}\\right)}\n\\right\\} \\\\\n&= \\langle {\\left(\\begin{smallmatrix}\n 0 & 1\\\\\n -1 & 0\n\\end{smallmatrix}\\right)}, {\\left(\\begin{smallmatrix}\n 1 & 1\\\\\n 1 & -1\n\\end{smallmatrix}\\right)}\\rangle \\subset {\\operatorname{SL}}\\_2({\\mathbb{F}}\\_3) \\subset M\\_2({\\mathbb{F}}\\_3)\n \\end{split}$$ \nis isomorphic to the group of quaternions. Another way to characterize *N*d, 3ʹ is\n \n$$\\label{eq:N3'p}\nN\\_{{\\mathrm{d}}, 3}' =\n\\left\\{\n\\pm\n\\begin{smat}\n { 1 }{ 0 }{ 0 }{ 1 }\n\\end{smat}\n\\right\\}\n\\cup\n\\left\\{\ng \\in {\\operatorname{SL}}\\_2({\\mathbb{F}}\\_3): {\\operatorname{Tr}}(g) = 0\n\\right\\}.$$ \nFrom this, it is easy to check the following local analog of Lemma [lemma:j] at 3.\n[lemma:j3] For any *j* ∈ Z/d3Z, we have \n*κ*d, 3(*T**j**N*d, 3ʹ) ∩ Ad, 30 = { ± [1, − *j*, 0, 1]}. \nDenote **0**3 ∈ A the zero matrix. Then *H*d, 3 acts on the set A ∖ **0**3, and decomposes it into 3 orbits according to the norm of the elements. The subgroup *H*d, 3ʹ ⊂ *H*d, 3 acts on A ∖ **0**3 similarly and decomposes the three orbits into 5 orbits. We denote the sum of elements in each orbit by w*i* for *i* = 0, 1, 2, 3, 4. They are explicitly given as follows:\n \n$$\\label{eq:wf}\n {\\mathfrak{w}}\\_i :=\n \\begin{cases}\n \\sum\\_{h \\in \\kappa\\_{{\\mathrm{d}}, 3}( T^{i}N\\_{{\\mathrm{d}}, 3}') } {\\mathfrak{e}}\\_h,& i= 0, 1, 2. \\\\\n \\sum\\_{h \\in {{\\mathcal{A}}}\\backslash {\\mathbf{0}}\\_3,~ \\det(h) \\equiv -i \\bmod{3} } {\\mathfrak{e}}\\_h,& i= 3, 4. \\\\\n \\end{cases}$$ \nThis gives *U*d, 3*H*d, 3ʹ ≅ Ce**0**3 + ∑*j* = 04Cw*j* ⊂ C[A]. Moreover, *U*d, 3*H*ʹd, 3 contains an SL2(Z)-invariant vector 4e**0**3 + w3, which is also in *U*d, 3*H*d, 3. Its orthogonal complement in *U*d, 3*H*ʹd, 3 is 5 dimensional and decomposes into $ \\chi\\_3^{-{\\mathrm{d}}\\_2} \\oplus \\chi\\_3^{-{\\mathrm{d}}\\_2} \\oplus \\varrho^{-{\\mathrm{d}}\\_2}$, where *χ*3 and $\\varrho\\_{}$ are irreducible representations of SL2(Z) given by\n \n$$\\label{eq:reps3}\n \\begin{split}\n \\chi\\_3(T) &= \\zeta^{}\\_3,~ \\chi\\_3(S) = 1, \\\\\n \\varrho(T) &=\n\\begin{pmatrix}\n1 & & \\\\\n& \\zeta\\_3 & \\\\\n& & \\zeta\\_3^2\n\\end{pmatrix},~\n\\varrho(S) =\n\\frac{1}{3} \\begin{pmatrix}\n -1 & 2 & 2 \\\\\n2 & -1 & 2 \\\\\n2 & 2 & -1\n\\end{pmatrix},\n \\end{split}$$ \nwith respect to the basis {w0 − w1, w0 − w2, 8e**0**3 − w3, w0 + w1 + w2, 2w4}. For any *m* ∈ Z, we use $\\varrho^{[m]}$ and *χ*3[*m*] to denote the representations of SL2(Z) defined by \n$$\\varrho^{[m]}(g) := \\varrho(g)^m,~\n\\chi\\_3^{[m]}(g) := \\chi\\_3(g)^m.$$ \n Note that $\\varrho^{[m]}$ and *χ*3[*m*] are well-defined and only depends on $m \\bmod 3$. We remark *χ*3∣Γ0(2) = *χ*8. In summary, we have\n1. The subrepresentation *ω*d, 3*H*d, 3 ⊂ *ω*d, 3 fixed by *H*d, 3 decomposes as \n$$\\omega\\_{{\\mathrm{d}}, 3}^{H\\_{{\\mathrm{d}}, 3}} \\cong \\mathds{1} \\oplus \\varrho^{[-{\\mathrm{d}}\\_2]}$$ \n with respect to the basis {4e**0**3 + w3, 8w0 − w3, w0 + w1 + w2, 2w4}.\n2. Denote *U*d, 3ʹ the orthogonal complement of *U*d, 3*H*d, 3 in *U*d, 3*H*d, 3ʹ and *ω*d, 3ʹ the restriction of *ω*d, 3 to *U*d, 3ʹ. Then \n*U*d, 3ʹ = {∑*j* = 02*a**j*w*j* : *a**j* ∈ C, ∑*j**a**j* = 0} \n and *ω*d, 3ʹ ≅ (*χ*3[ − d2])⊕ 2.\n3. Under this identification, $ M^{!}(\\omega\\_{{\\mathrm{d}}, 3})^{H\\_{{\\mathrm{d}}, 3}} \\cong M^! \\oplus M^!(\\varrho^{[-{\\mathrm{d}}\\_2]}) $ and \n*M*!(*ω*d, 3)*H*d, 3ʹ ≅ *M*!(*ω*d, 3)*H*d, 3 ⊕ *M*!(*χ*3[ − d2])⊕ 2.\nThe analog of ud satisfying Lemma [lemma:ufinv] and Proposition [prop:udinv] is in the subspace *U*d, 3ʹ = {ud, 3(*c⃗*) : *c⃗* = (*c**s*) ∈ C*φ*(d3)}, where\n \nud, 3(*c⃗*) : = ∑*j* ∈ Z/d3Z(∑*s* ∈ (Z/d3Z)×*c**s**ζ*d3*s**j*)(∑*h* ∈ *κ*d, 3(*T**j**N*d, 3ʹ)e*h*). \nAs a consequence of Lemma [lemma:ufinv], we have the following local analog of Prop. [prop:udinv] at *p* = 3.\n[prop:udinv3] For any d ∣ 24 and *c⃗* ∈ C*φ*(d3), we have\n \n*ω*d, 3(*g*)ud, 3(*c⃗*) = *χ*(*g*)− 24d2/d3ud, 3(*c⃗*) \nfor all *g* ∈ Γ0(2).\nIf d3 = 1, this is clear. Otherwise, \n*ω*d, 3(*T*)ud, 3(*c⃗*) = *χ*3− d2(*g*)ud, 3(*c⃗*) = *χ*− 8d2(*g*)ud, 3(*c⃗*). \n This finishes the proof.\nIf *c⃗* = (1, …, 1) ∈ C*φ*(d3), then we simply denote ud, 3(*c⃗*) by ud, 3, which is explicitly given by\n \n$$\\label{eq:uf3}\n {\\mathfrak{u}}\\_{{\\mathrm{d}}, 3} =\n \\begin{cases}\n 2{\\mathfrak{w}}\\_0 - {\\mathfrak{w}}\\_1 - {\\mathfrak{w}}\\_2,& {\\mathrm{d}}\\_3 = 3,\\\\\n {\\mathfrak{e}}\\_{{\\mathbf{0}}\\_3},& {\\mathrm{d}}\\_3 = 1.\n \\end{cases}$$ \nThe case *p* = 2\n----------------\nIn this case, the finite quadratic module Ad, 2 = Z/d2Z × Z/(2d2)Z × Z/(2d2)Z × Z/d2Z has the quadratic form\n \n$$\\label{eq:Q2}\nQ\\_{{\\mathrm{d}}, 2}([h\\_0, h\\_1, h\\_2, h\\_3]):= \\frac{{\\mathrm{d}}\\_3^{-1}}{2{\\mathrm{d}}\\_2} (2 h\\_0 h\\_3 - h\\_1 h\\_2) \\in \\frac{1}{2{\\mathrm{d}}\\_2}{\\mathbb{Z}}/{\\mathbb{Z}}.$$ \nEven though the size of Ad, 2 can be large, the number of orbits under the suitable orthogonal group *H*ʹd, 2 is much smaller. More precisely, we have *H*d, 2 = *N*d, 2 × *N*d, 2 and *H*ʹd, 2 ⊃ *N*ʹd, 2 × *N*ʹd, 2, where\n \n$$\\label{eq:H2}\n \\begin{split}\n N\\_{{\\mathrm{d}}, 2} &:= \\left\\{{\\begin{pmatrix}\n a & b\\\\\n 2c & d\n\\end{pmatrix}} \\in {\\operatorname{SL}}\\_2({\\mathbb{Z}}/(2{\\mathrm{d}}\\_2 {\\mathbb{Z}}))\\right\\}/\\langle T^{{\\mathrm{d}}\\_2}, C^{ {\\mathrm{d}}\\_2/(2, {\\mathrm{d}}\\_2)}\\rangle, \\\\\n N'\\_{{\\mathrm{d}}, 2} &:= \\langle A, C, D\\rangle \\cong ({\\mathbb{Z}}/({\\mathrm{d}}\\_2(2, {\\mathrm{d}}\\_2)/(4, {\\mathrm{d}}\\_2)){\\mathbb{Z}}\\times {\\mathbb{Z}}/({\\mathrm{d}}\\_2/(2, {\\mathrm{d}}\\_2)){\\mathbb{Z}}) \\rtimes {\\mathbb{Z}}/(4, {\\mathrm{d}}\\_2){\\mathbb{Z}}.\n \\end{split}$$ \nHere $ A:= {\\left(\\begin{smallmatrix}\n 3 & 2\\\\\n 4 & 3\n\\end{smallmatrix}\\right)}, C:= {\\left(\\begin{smallmatrix}\n 5 & 4\\\\\n 16 & 13\n\\end{smallmatrix}\\right)}, D := {\\left(\\begin{smallmatrix}\n -1 & 1\\\\\n -2 & 1\n\\end{smallmatrix}\\right)}$ are elements in SL2(Z) projected into *N*d, 2. The commutation relation is given by *D**A**D*− 1 = *A*3. In particular *N*ʹd, 2 has size d22 and is abelian for d2 = 1, 2, 4.\nThe group *N*d, 2 acts on the left on Ad, 2 via (simply coming from matrix multiplication)\n \n$$\\label{eq:N2act}\n {\\begin{pmatrix}\n a & b\\\\\n 2c & d\n\\end{pmatrix}} \\cdot [h\\_0, h\\_1, h\\_2, h\\_3] :=\n[ah\\_0 + bh\\_2, ah\\_1 + 2(bh\\_3), 2(ch\\_0) + dh\\_2, ch\\_1 + dh\\_3]$$ \nfor ${\\begin{pmatrix}\n a & b\\\\\n 2c & d\n\\end{pmatrix}} \\in N\\_{{\\mathrm{d}}, 2}$ and [*h*0, *h*1, *h*2, *h*3] ∈ Ad, 2. The same holds for the right action. We can embed *N*d, 2 into Ad, 2 using the map *κ*d, 2 : *N*d, 2 → Ad, 2 defined by\n \n$$\\label{eq:kappa2}\n \\kappa\\_{{\\mathrm{d}}, 2}{\\left (}{\\begin{pmatrix}\n a & b\\\\\n 2c & d\n\\end{pmatrix}} {\\right )}:= {\\mathrm{d}}\\_3^{-1} [a\\bmod{{\\mathrm{d}}\\_2}, 2b, 2c, d \\bmod{{\\mathrm{d}}\\_2}].$$ \nIt is then easy to check that\n \n$$\\label{eq:easy}\n\\begin{split}\n\\kappa\\_{{\\mathrm{d}}, 2}(g\\_1 g\\_2) &= g\\_1 \\cdot \\kappa\\_{{\\mathrm{d}}, 2}(g\\_2) = \\kappa\\_{{\\mathrm{d}}, 2}(g\\_1) \\cdot g\\_2,\n\\\\\n Q\\_{{\\mathrm{d}}, 2}(\\kappa\\_{{\\mathrm{d}}, 2}(g)) &= \\frac{{\\mathrm{d}}\\_3^{-1} \\det(g) \\bmod{{\\mathrm{d}}\\_2}}{{\\mathrm{d}}\\_2} \\in \\frac{2}{2{\\mathrm{d}}\\_2}{\\mathbb{Z}}/{\\mathbb{Z}}\\end{split}$$ \nfor all *g*, *g*1, *g*2 ∈ *N*d, 2. From this, it is easy to check that when 2 ∣ d, *κ*d, 2 is a 2-1 map since $({\\mathrm{d}}\\_2 + 1) {\\left(\\begin{smallmatrix}\n 1 & \\\\\n & 1\n\\end{smallmatrix}\\right)} \\in N\\_{{\\mathrm{d}}, 2}'$ and \n$$\\kappa\\_{{\\mathrm{d}}, 2} {\\left (}({\\mathrm{d}}\\_2 + 1) {\\left(\\begin{smallmatrix}\n 1 & \\\\\n & 1\n\\end{smallmatrix}\\right)} {\\right )}=\n\\kappa\\_{{\\mathrm{d}}, 2} {\\left (}{\\left(\\begin{smallmatrix}\n 1 & \\\\\n & 1\n\\end{smallmatrix}\\right)} {\\right )}.$$ \nTo better describe *κ*d, 2(*N*d, 2), it is useful to know the smallest additive subgroup of Ad, 2 containing it. We describe it in the following lemma.\n[lemma:additive] Let Ad, 2ʹ ⊂ Ad, 2 be the smallest (additive) subgroup containing *κ*d, 2(*N*ʹd, 2).\n1. When 8 ∣ d, Ad, 2ʹ ≅ (Z/2Z)2 × (Z/8Z)2 is the orthogonal complement of the subgroup generated by [6, 4, 0, 2], [0, 8, 0, 0], [0, 2, 2, 0] ∈ Ad, 2, and \n*κ*d, 2− 1(Ad, 2ʹ) = *N*ʹd, 2 ⊔ *T*4*N*ʹd, 2. \n Furthermore, we can distinguish the elements in *N*d, 2ʹ and *T*4*N*d, 2ʹ via\n \n$$\\label{eq:distinguish}\n \\begin{split}\n N\\_{{\\mathrm{d}}, 2}' &= \\kappa\\_{{\\mathrm{d}}, 2}^{-1} {\\left (}\\{[h\\_0, h\\_1, h\\_2, h\\_3] \\in {{\\mathcal{A}}}'\\_{{\\mathrm{d}}, 2}: h\\_0^2 - {\\mathrm{d}}\\_3^2 \\equiv h\\_1 + h\\_2 \\bmod{16} \\} {\\right )},\\\\\n T^4 N\\_{{\\mathrm{d}}, 2}' &= \\kappa\\_{{\\mathrm{d}}, 2}^{-1} {\\left (}\\{[h\\_0, h\\_1, h\\_2, h\\_3] \\in {{\\mathcal{A}}}'\\_{{\\mathrm{d}}, 2}: h\\_0^2 - {\\mathrm{d}}\\_3^2 \\equiv h\\_1 + h\\_2 + 8 \\bmod{16} \\} {\\right )}.\n \\end{split}$$\n2. When $8 \\nmid {\\mathrm{d}}$, Ad, 2ʹ ≅ (Z/d2Z)2 is generated by $\\kappa\\_{{\\mathrm{d}}, 2}({\\left(\\begin{smallmatrix}\n 1 & \\\\\n & 1\n\\end{smallmatrix}\\right)}), \\kappa\\_{{\\mathrm{d}}, 2}( {\\left(\\begin{smallmatrix}\n -1 & 1\\\\\n -2 & 1\n\\end{smallmatrix}\\right)})$ and \n*κ*d, 2− 1(Ad, 2ʹ) = *N*ʹd, 2.\nThis can be verified using the appendix and some computer calculation.\nIn addition, we record the following local analog of Lemma [lemma:j] at the prime 2.\n[lemma:j2] For any *j* ∈ Z/d2Z, we have \n*κ*d, 2(*T**j**N*d, 2ʹ) ∩ Ad, 20 = {[*r*, *r*(2*j* + (d3*r*)2 − 1), 0, *r*] : *r* ∈ (Z/d2Z)×}. \nFor *j* = 0, this follows directly from Lemma [lemma:additive]. In general, it is easy to check that \n*T**j*(*κ*d, 2(*N*d, 2ʹ) ∩ Ad, 20) = *κ*d, 2(*T**j**N*d, 2ʹ) ∩ Ad, 20 \n for any *j* since the action of *T* preserves Ad0.\nSince *T*d2 = 0 ∈ *N*ʹd, 2 and *H*d, 2ʹ is generated by *N*d, 2ʹ × *N*d, 2ʹ and (*T*, *T*), the index of *H*ʹd, 2 in *H*d, 2 is d2 and the sizes of *H*d, 2 and *H*ʹd, 2 are d26/(2, d2) and d25/(2, d2) respectively. The dimension of *U*d, 2*H*d, 2ʹ is the number of orbits in Ad, 2 under the action of *H*d, 2ʹ. Since the finite group *H*d, 2ʹ is explicitly given in, it is straightforward to calculate these orbits on a computer in practice. We did this in SAGE and received the following results\n \n$$\\label{eq:rd2}\n\\dim U\\_{{\\mathrm{d}}, 2}^{H\\_{{\\mathrm{d}}, 2}'} =\n \\begin{cases}\n 4& {\\mathrm{d}}\\_2 = 1,\\\\\n 16& {\\mathrm{d}}\\_2 = 2,\\\\\n 46& {\\mathrm{d}}\\_2 = 4,\\\\\n 118& {\\mathrm{d}}\\_2 = 8.\\\\\n \\end{cases}$$ \nWith these calculations, one can already explicitly decompose the representation *ω*d, 2*H*d, 2ʹ on *U*d, 2*H*d, 2ʹ. To find the desired vectors, we need to consider the following subspace of *U*d, 2*H*d, 2ʹ.\nFor d2 = 1, the vector e(1/2, 0) − e(0, 1/2) generates a 1-dimensional SL2(Z)-invariant subspace. Denote *U*d, 2ʹ ⊂ *U*d, 2*H*d, 2ʹ its orthogonal complement. For d2 ≥ 2, the subgroup *H*d, 2ʹ has index 2 in *H*d/2, 2ʹ = ⟨*T*d2/2, *H*d, 2ʹ⟩. Even though *U*d, 2*H*d, 2ʹ is large, it is much smaller compare to *U*d, 2 and is amenable to calculations on a computer. Denote *U*d, 2ʹ ⊂ *U*d, 2*H*d, 2ʹ the orthogonal complement of *U*d, 2*H*d/2, 2ʹ ⊂ *U*d, 2*H*d, 2ʹ. Then it is clear that \n$$\\dim U\\_{{\\mathrm{d}}, 2}' = \\frac{1}{2} {\\left (}\\text{numbers of } H\\_{{\\mathrm{d}}, 2}'\\text{-orbits of } {{\\mathcal{A}}}\\_{{\\mathrm{d}}, 2}\n- \\text{numbers of } H\\_{{\\mathrm{d}}/2, 2}'\\text{-orbits of } {{\\mathcal{A}}}\\_{{\\mathrm{d}}, 2}\n{\\right )}.$$ \n The following result comes out of the computer calculations.\n[lemma:computer] For any d ∣ 24, the dimension of *U*d, 2ʹ is 3*φ*(d2). Furthermore, the support of any elements in *U*d, 2ʹ is contained in the union of $\\{h \\in {{\\mathcal{A}}}\\_{{\\mathrm{d}}, 2}: 2Q\\_{\\mathrm{d}}(h) = -\\frac{{\\mathrm{d}}\\_3^{-1}}{{\\mathrm{d}}\\_2}\\}$ and ∪ *j* ∈ Z/d2Z*κ*d, 2(*T**j**N*d, 2ʹ).\nNow for d ∣ 24, define the following vectors\n \n$$\\label{eq:ud2c}\n \\begin{split}\n {\\mathfrak{u}}\\_{{\\mathrm{d}}, 2}(\\vec{c}) &:= \\sum\\_{j \\in {\\mathbb{Z}}/{\\mathrm{d}}\\_2 {\\mathbb{Z}}}\n {\\left (}\\sum\\_{s \\in ({\\mathbb{Z}}/{\\mathrm{d}}\\_2{\\mathbb{Z}})^\\times} c\\_s \\zeta\\_{{\\mathrm{d}}\\_2}^{js}{\\right )}{\\left (}\\sum\\_{h \\in \\kappa\\_{{\\mathrm{d}}, 2}(T^j N\\_{{\\mathrm{d}}, 2}')} {\\mathfrak{e}}\\_h {\\right )}, \\\\\n {\\mathfrak{v}}\\_{{\\mathrm{d}}, 2}(\\vec{c}) &:= \\omega\\_{{\\mathrm{d}}, 2}(S) {\\mathfrak{u}}\\_{{\\mathrm{d}}, 2}(\\vec{c}),~\n {\\mathfrak{w}}\\_{{\\mathrm{d}}, 2}(\\vec{c}) := \\zeta^{-{\\mathrm{d}}\\_3}\\_{{\\mathrm{d}}\\_2} \\omega\\_{{\\mathrm{d}}, 2}(T) {\\mathfrak{v}}\\_{{\\mathrm{d}}, 2}(\\vec{c})\n \\end{split}$$ \nfor all *c⃗* = (*c**s*) ∈ C*φ*(d2). From Lemma [lemma:computer], we can show that these vectors give a basis of *U*d, 2ʹ.\n[lemma:support] For any d ∣ 24 with 2 ∣ d and *c⃗* ∈ C*φ*(d2), the vectors vd, 2(*c⃗*), wd, 2(*c⃗*) have the same support, which is disjoint from that of ${\\mathfrak{u}}\\_{{\\mathrm{d}}, 2}(\\vec{c\\_1})$ for any $\\vec{c\\_1} \\in {\\mathbb{C}}^{\\varphi({\\mathrm{d}}\\_2)}$.\nSince the action of *ω*d, 2(*T*) does not change the support, we know that vd, 2(*c⃗*) and wd, 2(*c⃗*) have the same support. Now we have by definition \nvd, 2(*c⃗*) = (2d22)− 1∑*s* ∈ (Z/d2Z)×*c**s*∑*μ* ∈ Ad, 2∑*j* ∈ Z/d2Z*ζ*d2*j**s*(∑*h* ∈ *κ*d, 2(*T**j**N*d, 2ʹ)**e**((*μ*, *h*))e*μ*). \n We want to show that the coefficient of e*μ* is zero if *μ* = *κ*d, 2(*T**j*ʹ*g*ʹ) with *j*ʹ ∈ Z/d2Z and *g*ʹ ∈ *N**d*, 2ʹ. Now if *h* = *κ*d, 2(*g*) with *g* ∈ *T**j**N**d*, 2ʹ, then $(\\mu, h) = \\frac{{\\mathrm{d}}\\_3^{-1}{\\operatorname{Tr}}(g (T^{j'}g')^{-1})}{{\\mathrm{d}}\\_2}$ by. Since *N*d, 2ʹ is normal in *N*d, 2, which contains *T*, we have *T**j**N*d, 2ʹ*T*− *j*ʹ = *T**j* − *j*ʹ*N*d, 2ʹ. Therefore, it suffices to show that the sum below vanishes\n \n$$\\begin{aligned}\n \\sum\\_{j \\in {\\mathbb{Z}}/{\\mathrm{d}}\\_2{\\mathbb{Z}}}\n& \\zeta\\_{{\\mathrm{d}}\\_2}^{js}\n \\sum\\_{h \\in \\kappa\\_{{\\mathrm{d}}, 2}(T^j N\\_{{\\mathrm{d}}, 2}')} {{\\mathbf{e}}}((\\mu, h))\n=\n\\zeta\\_{{\\mathrm{d}}\\_2}^{j's} \\sum\\_{j'' \\in {\\mathbb{Z}}/{\\mathrm{d}}\\_2{\\mathbb{Z}}}\n \\zeta\\_{{\\mathrm{d}}\\_2}^{j''s}\n \\sum\\_{h \\in \\kappa\\_{{\\mathrm{d}}, 2}(T^{j''} N\\_{{\\mathrm{d}}, 2}')} \\zeta\\_{{\\mathrm{d}}\\_2}^{{\\mathrm{d}}\\_3^{-1} {\\operatorname{Tr}}(h)}\\end{aligned}$$ \nwith *j*ʺ : = *j* − *j*ʹ. Also for *h* = [*h*0, *h*1, *h*2, *h*3] ∈ *κ*d, 2(*N*d, 2ʹ), we have Tr(*T**j* ⋅ *h*) = Tr(*h*) + *j* ⋅ *h*1. Using this, we can rewrite\n \n$$\\begin{aligned}\n \\sum\\_{j'' \\in {\\mathbb{Z}}/{\\mathrm{d}}\\_2{\\mathbb{Z}}}\n& \\zeta\\_{{\\mathrm{d}}\\_2}^{j''s}\n \\sum\\_{h \\in \\kappa\\_{{\\mathrm{d}}, 2}(T^{j''} N\\_{{\\mathrm{d}}, 2}')} \\zeta\\_{{\\mathrm{d}}\\_2}^{{\\mathrm{d}}\\_3^{-1} {\\operatorname{Tr}}(h)}\n=\n \\sum\\_{h \\in \\kappa\\_{{\\mathrm{d}}, 2}(N\\_{{\\mathrm{d}}, 2}')}\n\\zeta\\_{{\\mathrm{d}}\\_2}^{{\\mathrm{d}}\\_3^{-1} {\\operatorname{Tr}}(h)}\n\\sum\\_{j'' \\in {\\mathbb{Z}}/{\\mathrm{d}}\\_2{\\mathbb{Z}}}\n \\zeta\\_{{\\mathrm{d}}\\_2}^{j''(s + {\\mathrm{d}}\\_3^{-1} h\\_1)}\\end{aligned}$$ \nBy Lemma [lemma:additive] (or inspecting the appendix), we know that *h*1 ∈ 2Z/2d2Z for all *h* ∈ *κ*d, 2(*N*d, 2ʹ). So *s* + d3− 1*h*1 ∈ (Z/d2Z)× and the sum above vanishes.\n[lemma:basis] For any d ∣ 24 and any basis B of C*φ*(d2), the set\n \n⋃*c⃗* ∈ B{ud, 2(*c⃗*), vd, 2(*c⃗*), wd, 2(*c⃗*)} \nis a basis of *U*d, 2ʹ.\nWe know that dimension of *U*ʹd, 2 is 3*φ*(d2) from Lemma [lemma:computer], and need to check linear independence of the vectors in the set above. Since the vectors ud, 2(*c⃗*), vd, 2(*c⃗*), wd, 2(*c⃗*) are defined linearly, it suffices to prove the lemma for B = {*e⃗*(*s*0) : *s*0 ∈ (Z/d2Z)×} with *e⃗*(*s*0) ∈ C*φ*(d2) the standard basis vector with 0 everywhere except 1 at the *s*0th entry. It is easily checked from the definition that ud, 2(*c⃗*), vd, 2(*c⃗*), wd, 2(*c⃗*) are in *U*d, 2ʹ are eigenvectors of *T* with eigenvalue *ζ*d2− *s*0 when *c⃗* = *e⃗*(*s*0). Therefore, it suffices to check that the three vectors ud, 2(*c⃗*), vd, 2(*c⃗*), wd, 2(*c⃗*) are linearly independent whenever *c⃗* = *e⃗*(*s*0).\nWhen d2 = 1, this is easily checked by hand. When d2 ≥ 2, it suffices to show that vd, 2(*e⃗*(*s*0)) and wd, 2(*e⃗*(*s*0)) are linearly independent by Lemma [lemma:support]. Assume otherwise, then the restriction of *ω*d, 2 to Cud, 2(*e⃗*(*s*0)) + Cvd, 2(*e⃗*(*s*0)) is a 2-dimensional representation of SL2(Z). In the basis {ud, 2(*e⃗*(*s*0)), vd, 2(*e⃗*(*s*0))}, it is given by the map \n$$T \\mapsto {\\begin{pmatrix}\n \\zeta\\_{{\\mathrm{d}}\\_2}^{-1} & \\\\\n & \\pm \\zeta\\_{2{\\mathrm{d}}\\_2}\n\\end{pmatrix}},~\nS \\mapsto {\\begin{pmatrix}\n & 1\\\\\n 1 & \n\\end{pmatrix}}.$$ \n However, (*T* ⋅ *S*)6 is the identity, whereas \n$${\\left (}{\\begin{pmatrix}\n \\zeta\\_{{\\mathrm{d}}\\_2}^{-1} & \\\\\n & \\pm \\zeta\\_{2{\\mathrm{d}}\\_2}\n\\end{pmatrix}}\n{\\begin{pmatrix}\n & 1\\\\\n 1 & \n\\end{pmatrix}} {\\right )}^6\n= {\\left (}\\pm {\\begin{pmatrix}\n \\zeta\\_{2{\\mathrm{d}}\\_2}^{-1} & \\\\\n & \\zeta\\_{2{\\mathrm{d}}\\_2}^{-1}\n\\end{pmatrix}} {\\right )}^3$$ \n is not the identity since 2 ∣ d2. This is a contradiction and finishes the proof.\n[prop:decomp2] For d ∣ 24, let *ω*d, 2ʹ denote the restriction of *ω*d, 2 to *U*d, 2ʹ ⊂ *U*d, 2*H*d, 2ʹ. Then ud, 2(*c⃗*) satisfies\n \n*ω*d, 2ʹ(*g*)ud, 2(*c⃗*) = *χ*(*g*)− 24d3/d2ud, 2(*c⃗*) \nfor all *g* ∈ Γ0(2) and *c⃗* ∈ C*φ*(d2). Furthermore with respect to the basis in, we have\n \n$$\\label{eq:decomp2}\n \\omega\\_{{\\mathrm{d}}, 2}' \\cong\n \\varrho\\_{{\\mathrm{d}}\\_2, {\\mathrm{d}}\\_3}^{\\oplus \\varphi({\\mathrm{d}}\\_2)}$$ \nHere $\\varrho\\_{{\\mathrm{d}}\\_2, {\\mathrm{d}}\\_3}$ is the 3-dimensional representation defined in.\nIf *c⃗* = (1, …, 1) ∈ C*φ*(d2), we will simply write ud, 2 for ud, 2(*c⃗*). They are explicitly given by\n \n$$\\label{eq:uf2}\n{\\mathfrak{u}}\\_{{\\mathrm{d}}, 2} =\n \\begin{cases}\n{\\mathfrak{e}}\\_{{\\mathbf{0}}\\_2}, & {\\mathrm{d}}\\_2 = 1, \\\\\n2^{{\\mathrm{d}}\\_2/2} {\\left (}\\sum\\_{\\kappa\\_{{\\mathrm{d}}, 2}(N\\_{{\\mathrm{d}}, 2}')} {\\mathfrak{e}}\\_h - \\sum\\_{\\kappa\\_{{\\mathrm{d}}, 2}(T^{{\\mathrm{d}}\\_2/2} N\\_{{\\mathrm{d}}, 2}' )} {\\mathfrak{e}}\\_h {\\right )}, & {\\mathrm{d}}\\_2 = 2, 4, 8.\n \\end{cases}$$ \nFor the first claim, it suffices to prove the it when *g* = *T*, *S*2, *T**B*, which are generators of Γ0(2). If *g* = *T*, then *ω*d, 2ʹ(*T*)e*h* = **e**( − *Q*d, 2(*h*))e*h*. For *h* ∈ *κ*d, 2(*T**j**N*d, 2ʹ), we have $Q\\_{{\\mathrm{d}}, 2}(h) = {\\mathrm{d}}\\_3^{-1}/{\\mathrm{d}}\\_2 = {\\mathrm{d}}\\_3/{\\mathrm{d}}\\_2 \\in \\tfrac{1}{{\\mathrm{d}}\\_2} {\\mathbb{Z}}/{\\mathbb{Z}}$. Equation therefore holds for *g* = *T*. When *g* = *S*2, since *ω*d(*S*2)e*h* = e− *h* for all *h* ∈ Ad and $-{\\left(\\begin{smallmatrix}\n 1 & \\\\\n & 1\n\\end{smallmatrix}\\right)} \\in N\\_{{\\mathrm{d}}, 2}'$, we know that − *κ*d(*T**j**N*d, 2ʹ) = *κ*d(*T**j**N*d, 2ʹ) and equation holds for *g* = *S*2.\nFor *g* = *T**B* = *T**S**T*2*S*− 1, it suffices to show that the middle equation below \n*ω*d, 2ʹ(*S*)*ω*d, 2(*T*− 1)ud, 2(*c⃗*) = *ζ*d2d3− 1vd, 2(*c⃗*) = *ω*d, 2(*T*2)vd, 2(*c⃗*) = *ω*d, 2(*T*2)*ω*d, 2ʹ(*S*)ud, 2(*c⃗*). \n This is easily checked by hand when d2 = 1. If 2 ∣ d2, we know by Lemmas [lemma:computer] and [lemma:support] that the support of vd, 2(*c⃗*) is contained in $\\{h \\in {{\\mathcal{A}}}\\_{{\\mathrm{d}}, 2}: 2Q\\_{\\mathrm{d}}(h) = - \\frac{{\\mathrm{d}}\\_3^{-1}}{{\\mathrm{d}}\\_2} \\}$. It is therefore an eigenvector of *ω*ʹd, 2(*T*2) with eigenvalue *ζ*d2d3− 1. This proves the first claim. As in the proof of Prop. [prop:repembed], the vectors {ud, 2(*c⃗*), vd, 2(*c⃗*), wd, 2(*c⃗*)} generate a 3-dimensional subrepresentation of *ω*d, 2 isomorphic to $\\varrho\\_{{\\mathrm{d}}\\_2, {\\mathrm{d}}\\_3}$. The second claim then follows from Lemma [lemma:basis].\nBorcherds liftings\n==================\nBrief review of Borcherds liftings\n----------------------------------\nWe first set up notation and briefly review the Borcherds lifting, following. Let *V* = *V*d and *H* be as in Section [sect3].\nLet *V* = *M*2(Q) with quadratic form *Q*(*x*) = ddet*x* and let *L* = *L*d be as in section [subsec:Lattice]. Then \n*H* = GSpin(*V*) = {*g* = (*g*1, *g*2) ∈ GL2 × GL2 : det*g*1 = det*g*2} \n acting on *V* via \n(*g*1, *g*2)*x* = *g*1*x**g*2− 1. \n One has the exact sequence \n$$1 \\rightarrow \\mathbb G\\_m \\rightarrow H \\rightarrow {{\\mathrm{SO}}}(V) \\rightarrow 1.$$ \nLet\n \n$$\\mathcal L = \\{ w \\in V\\_{\\mathbb{C}}:\\, (w, w) =0, \\quad (w, \\bar w)<0\\}.$$ \nand let $\\mathbb D$ be the Hermitian symmetric domain of oriented negative 2-planes in *V*R = *V* ⊗ QR. Then one has an isomorphism \n$$pr: \\mathcal L/{\\mathbb{C}}^\\times \\cong \\mathbb D, \\quad w=u + i v \\mapsto {\\mathbb{R}}u + {\\mathbb{R}}(-v).$$ \n For the isotropic matrix $\\ell= {\\left(\\begin{smallmatrix} 0 & -1 \\\\ 0 & 0\\end{smallmatrix}\\right) } \\in L$ and $\\ell' ={\\left(\\begin{smallmatrix} 0 & 0 \\\\ \\frac{1}{{\\mathrm{d}}} & 0\\end{smallmatrix}\\right) } \\in V$ with (ℓ, ℓʹ) = 1. We also have the associated tube domain \n$$\\mathcal H\\_{\\ell, \\ell'} =\\{ {\\left(\\begin{smallmatrix} z\\_1 & 0 \\\\ 0 & -z\\_2\\end{smallmatrix}\\right) } : \\, y\\_1 y\\_2 >0\\},\\quad y\\_i=\\hbox{Im}(z\\_i),$$ \n together with \n$$w: \\mathcal H\\_{\\ell, \\ell'} \\rightarrow \\mathcal L, \\quad w( {\\left(\\begin{smallmatrix} z\\_1 & 0 \\\\ 0 & -z\\_2\\end{smallmatrix}\\right) }) = {\\left(\\begin{smallmatrix} z\\_1 & -{\\mathrm{d}}z\\_1 z\\_2 \\\\ \\frac{1}{{\\mathrm{d}}} & -z\\_2\\end{smallmatrix}\\right) }.$$ \n This gives an isomorphism $\\mathcal H\\_{\\ell, \\ell'}\\cong \\mathcal L/{\\mathbb{C}}^\\times$. We also identity H2 ∪ (H−)2 with $\\mathcal H\\_{\\ell, \\ell'}$ by \n$$\\psi\\_{\\mathrm{d}}: z=(z\\_1,z\\_2)\\mapsto{\\left(\\begin{smallmatrix} \\frac{z\\_1}{{\\mathrm{d}}} & 0 \\\\ 0 & \\frac{z\\_2}{{\\mathrm{d}}}\\end{smallmatrix}\\right) }.$$ \n Note that we use this identification in order to have the following compatibility property and it is also the identification used in the computation of Borcherds products. The following is a special case of \nDefine \n$$w\\_{{\\mathrm{d}}}: \\, {\\mathbb{H}}^2 \\cup ({\\mathbb{H}}^-)^2 \\rightarrow \\mathcal L, \\quad w\\_{{\\mathrm{d}}}(z\\_1, z\\_2) = w\\circ\\psi\\_{\\mathrm{d}}(z\\_1,z\\_2)={\\left(\\begin{smallmatrix} \\frac{z\\_1}{{\\mathrm{d}}} & \\frac{-z\\_1 z\\_2}{{\\mathrm{d}}} \\\\ \\frac{1}{{\\mathrm{d}}} & \\frac{-z\\_2}{{\\mathrm{d}}}\\end{smallmatrix}\\right) }.$$ \n Then the composition *p**r* ∘ *w*d gives an isomorphism between H2 ∪ (H−)2 and $\\mathbb D$. Moreover, *w*d is *H*(R)-equivariant, where *H*(R) acts on H2 ∪ (H−)2 via the usual linear fraction: \n(*g*1, *g*2)(*z*1, *z*2) = (*g*1(*z*1), *g*2(*z*2)), \n and acts on $\\mathcal L$ and $\\mathbb D$ naturally via its action on *V*. Moreover, one has\n \n$$\\label{eq:linebundle}\n(g\\_1, g\\_2) w\\_{{\\mathrm{d}}}(z\\_1, z\\_2) = \\frac{j(g\\_1, z\\_1) j(g\\_2, z\\_2)}{\\nu(g\\_1, g\\_2)} w\\_{{\\mathrm{d}}}(g\\_1(z\\_1), g\\_2(z\\_2)),$$ \nwhere *ν*(*g*1, *g*2) = det*g*1 = det*g*2 is the spin character of *H* ≅ GSpin(*V*), and \n*j*(*g*1, *g*2, *z*1, *z*2) = *j*(*g*1, *z*1)*j*(*g*2, *z*2) = (*c*1*z*1 + *d*1)(*c*2*z*2 + *d*2) \n is the automorphy factor (of weight (1, 1)).\nFor a congruence subgroup Γ of SL2(Z), let *X*Γ be the associated open modular curve over Q such that *X*Γ(C) = Γ ∖ H. Assume Γ ⊃ Γ(*M*) for some integer *M* ≥ 1. Let \n$$\\nu: {\\mathbb{A}}^\\times \\hookrightarrow {{\\mathrm{GL}}}\\_2({\\mathbb{A}}), \\quad \\nu(d) = {\\operatorname{diag}}(1, d).$$ \n Let *K*(Γ) be the product of $\\nu(\\hat{{\\mathbb{Z}}}^\\times)$ and the preimage of Γ/Γ(*M*) in ${{\\mathrm{GL}}}\\_2(\\hat{\\mathbb{Z}})$ (under the map ${{\\mathrm{GL}}}\\_2(\\hat{\\mathbb{Z}})\\rightarrow {{\\mathrm{GL}}}\\_2({\\mathbb{Z}}/M))$. Let *K* = (*K*(Γ) × *K*(Γ)) ∩ *H*(A*f*). Then one has by the strong approximation theorem \n*X**K* ≅ *X*Γ × *X*Γ. \n In this way, we have identified the product of two copies of a modular curve *X*Γ with a Shimura variety *X**K*.\nSuppose that Γ acts on *L*ʹ/*L* trivially, then for each *μ* ∈ *L*ʹ/*L* and *m* ∈ *Q*(*μ*) + *L*, the associated special divisor *Z*Γ(*m*, *μ*) is given by \n$$Z\\_{\\Gamma}(m, \\mu) =(\\Gamma \\times \\Gamma)\\backslash \\{ (z\\_1, z\\_2):\\, w\\_{{\\mathrm{d}}}(z\\_1, z\\_2) \\perp x \\hbox{ for some } x \\in \\mu+L, Q(x)=m\\}.$$ \n More generally, assume Γ ⊃ Γ(*M*) preserves *L*, and u = ∑*a**μ*e*μ* ∈ C[*L*ʹ/*L*] is Γ × Γ-invariant, the cycle \n*Z*Γ(*M*)(*m*, u) = ∑*a**μ**Z*Γ(*M*)(*m*, *μ*) \n descends to a cycle *Z*Γ(*m*, u) in *X*Γ × *X*Γ. For our purpose, we will take \nd ∣ 24, Γ = Γ*χ*, d ⊃ Γd ⊃ Γ(2d) ⊃ Γ(48) \n from now on and write *X*d : = *X*Γ = *X*Γ*χ*, d. Notice that *X*1 = *X*0(2) has two cusps, *i*∞ and 0. Since {*T**j* : 1 ≤ *j* ≤ d} are coset representatives of Γ*χ*, d in Γ*χ*, 1, the modular curve *X*d has the same cusps as *X*1.\n[Corollary 3.3] [lemma:YYdivisor] For d ∣ dʹ ∣ 24, let *π* : *X*Γ(2dʹ) → *X*d be the natural projection. Then\n \n$$\\label{eq:diagonal\\*}\n(\\pi\\_{} \\times \\pi\\_{})^\\*(X\\_{\\mathrm{d}}^\\Delta) =\\sum\\_{\\gamma \\in \\Gamma\\_{}/\\Gamma(2{\\mathrm{d}}')} Z\\_{\\Gamma(2{\\mathrm{d}}')} {\\left (}\\frac{1}{{\\mathrm{d}}'}, \\frac{1}{{\\mathrm{d}}'}\\gamma +L{\\right )}$$ \nand the group Γ(2dʹ) can be replaced by Γdʹ.\nSince Γ is normal in Γ0(2) = Γ*χ*, 1, the action of $T = {\\left(\\begin{smallmatrix}\n 1 & 1\\\\\n 0 & 1\n\\end{smallmatrix}\\right)} \\in \\Gamma\\_0(2)$ on H factors through *X*d and defines a map *X*d → *X*d, which we also denote by *T*. Using this, we can define translates of the diagonal\n \n$$\\label{eq:diagonalj}\n X\\_{\\mathrm{d}}^\\Delta(j) := (T^j \\times \\mathds{1})^\\*(X\\_{\\mathrm{d}}^\\Delta) \\subset X\\_{\\mathrm{d}}\\times X\\_{\\mathrm{d}}$$ \nfor *j* ∈ Z/dZ. Equation also generalizes to\n \n$$\\label{eq:diagonalj\\*}\n(\\pi\\_{} \\times \\pi\\_{})^\\*(X\\_{\\mathrm{d}}^\\Delta(j)) =\\sum\\_{\\gamma \\in \\Gamma\\_{}/\\Gamma(2{\\mathrm{d}}')} Z\\_{\\Gamma(2{\\mathrm{d}}')} {\\left (}\\frac{1}{{\\mathrm{d}}'}, \\frac{1}{{\\mathrm{d}}'}T^j\\gamma +L{\\right )},$$ \nwhere one can replace Γ(2dʹ) with Γdʹ. From this, we see that the pull back of *X*dΔ(*j*) along natural projection *X*dʹ × *X*dʹ → *X*d × *X*d is ∪ *l* ∈ dZ/dʹZ*X*dʹΔ(*j* + *l*). Before proceeding further to state and prove the main result of this section, we record the following identity for convenience.\n[lemma:pol] For any d ∈ N, we have the following identity in Q(*X*)\n \npd(*X*) : = ∏*j* ∈ Z/dZ(1 − *ζ*d*j**X*)*a*d(*j*) = ∏*b* ∣ d(1 − *X*d/*b*)*b* ⋅ *μ*(d/*b*), \nwhere *a*d(*j*) is the constant defined in. Furthermore for any *s* ∈ N, we have\n \n∏d ∣ *s*pd(*X**s*/d) = (1 − *X*)*s*. \nTo prove, it suffices to check that both sides have the same roots counting multiplicity, since they agree at *X* = 0. The multiplicity of *X* = *ζ*d*j* on the LHS is *a*d( − *j*) = *a*d(*j*), whereas it is ∑*b* ∣ (d, *j*)*b* ⋅ *μ*(d/*b*) on the RHS. The equality is then a consequence of the identity \n$$\\sum\\_{b \\mid n} b\\cdot \\mu {\\left (}\\frac{{\\mathrm{d}}}{b} {\\right )}= \\mu({\\mathrm{d}}/n) \\frac{\\varphi({\\mathrm{d}})}{\\varphi({\\mathrm{d}}/n)},~ n \\mid {\\mathrm{d}},$$ \n which is a standard exercise that we leave, along with, to the curious readers.\nNow, we can specialize Borcherds’ far reaching lifting theorem ( (see also ) to the modular function *F̃*d in and the result below.\n[theo:BorcherdsLifting] For every d ∣ 24, recall the modular function in *M*!(*ω*d)*H*dʹ \n$$\\tilde{F}\\_{\\mathrm{d}}(\\tau) =\n \\sqrt{2}^{\\mathrm{d}}{\\left (}{\\mathfrak{f}}\\_2^{-24/{\\mathrm{d}}}(\\tau) {\\mathfrak{u}}\\_{\\mathrm{d}}+ {\\mathfrak{f}}\\_1^{-24/{\\mathrm{d}}}(\\tau) {\\mathfrak{v}}\\_{\\mathrm{d}}+ {\\mathfrak{f}}^{-24/{\\mathrm{d}}} (\\tau) {\\mathfrak{w}}\\_{\\mathrm{d}}{\\right )}+\n \\begin{cases}\n 24({\\mathfrak{e}}\\_{(0, 0)} + {\\mathfrak{e}}\\_{(1/2, 0)}),& {\\mathrm{d}}= 1,\\\\\n 0, & {\\mathrm{d}}> 1,\n \\end{cases}$$ \n defined in with ud, vd, wd ∈ *U*d vectors defined in and. Let Ψd(*z*) be the meromorphic modular function on *X*d × *X*d (with some characters) associated to *F̃*d via Borcherds multiplicative lifting, i.e. − log∥Ψd(*z*)∥Pet2 is the regularized theta lift of *F̃*d with ∥ ⋅ ∥Pet a suitably normalized Petersson norm (see e.g. Theorem 2.1 in ). Then Ψd(*z*) has the following properties:\n1. one has on *X*d × *X*d \nDiv(Ψd(*z*)) = ∑*j* ∈ Z/dZ*a*d(*j*)*X*dΔ(*j*)\n2. When d = 1, Ψd(*z*) has a product expansion of the form \nΨ1(*z*) = 212(*q*1 − *q*2)∏*m*, *n* ≥ 1(1 − *q*1*n**q*2*m*)*c*1(*m**n*)(1 − *q*12*n**q*22*m*)*c*− 1(2*m**n*) \n near the cusp Qℓ of *X**K*, where *q**j* : = *e*2*π**i**z**j* and *c*d(*l*) are the Fourier coefficients defined in Remark [rmk:FC].\n3. When d > 1, Ψd(*z*) has a product expansion of the form \n$$\\Psi\\_{\\mathrm{d}}(z)=\n \\prod\\_{b \\mid {\\mathrm{d}}} (q\\_1^{1/b} - q\\_2^{1/b})^{b \\cdot \\mu({\\mathrm{d}}/b)}\n\\prod\\_{\\substack{m, n \\in {\\mathbb{N}}\\\\ mn \\equiv -1 \\bmod{{\\mathrm{d}}}}}\n {\\left (}\\prod\\_{b \\mid {\\mathrm{d}}} (1 - q\\_1^{n/b}q\\_2^{m/b})^{b \\cdot \\mu({\\mathrm{d}}/b)}\n{\\right )}^{c\\_{\\mathrm{d}}(mn)(-1)^{(n^2 - 1)/{\\mathrm{d}}\\_2}}$$ \n near the cusp Qℓ of *X**K*, where *μ* and *φ* are the Möbius and Euler *φ*-function respectively.\nThis is a specialization of Borcherds’ result to the input *F̃*d ∈ *M*!(*ω*d). For this, we need to substitute the suitable parameters into Borcherds’ result, which has been specialized to this case in Theorem 2.1 and 2.2 in. Using the specialization there, we see that the divisor of Ψd is\n \n$$\\begin{aligned}\n \\sum\\_{j \\in {\\mathbb{Z}}/{\\mathrm{d}}{\\mathbb{Z}}} a\\_{\\mathrm{d}}(j)\\sum\\_{\\mu \\in \\kappa(T^jN\\_{\\mathrm{d}}')} Z\\_{\\Gamma\\_{\\mathrm{d}}}{\\left (}-\\frac{1}{{\\mathrm{d}}}, \\mu {\\right )}&=\n\\sum\\_{j \\in {\\mathbb{Z}}/{\\mathrm{d}}{\\mathbb{Z}}} a\\_{\\mathrm{d}}(j) \\sum\\_{\\gamma \\in \\Gamma /\\Gamma(2{\\mathrm{d}})} Z\\_{\\Gamma(2{\\mathrm{d}})} {\\left (}- \\frac{1}{{\\mathrm{d}}}, \\frac{1}{{\\mathrm{d}}} T^j \\gamma + L{\\right )},\n \\end{aligned}$$ \nwhich gives us the first claim after applying Lemma [lemma:YYdivisor].\nFor the second and third claim, we specialize Theorem 2.2 in and use the notations there. When d = 1, this is rather classical and can be found in (see also Prop. 5.3 in [3](#fn3)). For d > 1, the Weyl chambers for *F̃*d are the same as in the case d = 1, and we choose the one $W = {\\mathbb{R}}\\{{\\left(\\begin{smallmatrix}\n a & \\\\\n & -1\n\\end{smallmatrix}\\right)}: a > 1\\}$. Since ud, vd and wd do not have support on any isotropic vector, the associated form *F̃*d, *P* is identically zero, and the"},"data_id":{"kind":"string","value":"arxiv_0185574"}}},{"rowIdx":46398,"cells":{"content":{"kind":"string","value":"5) and (-0.2,1.8).. (0,1.8).. controls (0.2,1.8) and (0.3,1).. (0.3,1.2) ; (-2.3,1.1).. controls (-2.3,2) and (-1.55,2).. (-1.45,1.75).. controls (-1.35,1.25) and (-1.7,1.5).. (-1.7,1).. controls (-1.7,-0.2) and (0,-0.8).. (0.5,-0.8).. controls (2,-0.8) and (1.8,0).. (1.3,1.3).. controls (1.3,1.8).. (2,1.8).. controls (2.6,1.8).. (2.6,1.1) ;\n[baseline=(cdots.base)] (-2.5,1)–(-1.25,1)–(-1.25,2)–(-2.5,2)–cycle; at (-1.875,1.5) *p*1; (-0.75,1)–(0.5,1)–(0.5,2)–(-0.75,2)–cycle; at (-0.125,1.5) *p*2; (cutPR) at (-0.5,-0.25) cut; (cutQR) at (1.25,-0.5) cut; (-1.875,1) to[out=-90,in=180] (cutPR); (-0.125,1) to[out=-90,in=180] (cutQR); (1,1)–(3,1)–(3,2)–(1,2)–cycle; at (2,1.5) *q*; (1.5,1) to[out=-90,in=0] (cutPR); (2.5,1) to[out=-90,in=0] (cutQR); (-2.3,1.1).. controls (-2.3,2) and (-1.55,2).. (-1.45,1.75).. controls (-1.35,1.25) and (-1.7,1.5).. (-1.7,1).. controls (-1.7,-0.2) and (-1,-0.6).. (-0.5,-0.6).. controls (0.5,-0.6) and (1.3,0).. (1.3,1.3).. controls (1.3,1.8).. (2,1.8).. controls (2.6,1.8).. (2.6,1).. controls (2.6,-0.5) and (2,-0.85).. (1.25,-0.85).. controls (0.5,-0.85) and (-0.4,0).. (-0.4,1.1).. controls (-0.4,1.5) and (-0.2,1.8).. (0,1.8).. controls (0.2,1.8) and (0.3,1).. (0.3,1.2) ;\n[fig:slipknot]\nIn the remainder of this section, we fix a reduction step $p\\totom q$, and we show that the previous example describes a general mechanism: a path *χ* in *q* that is not already in *p* must involve a subpath *χ*ʹ between two residuals of a cut of *p* that was eliminated in $p\\totom q$. We refer to this situation as a in *χ*.\nMore formally, consider $c=\\cut{t\\_0}{s\\_0}\\in \\cuts(p)$ with $t\\_0=\\otimes(t\\_1,\\dotsc,t\\_n)$ and $s\\_0=\\parr(s\\_1,\\dotsc,s\\_n)$ and assume *c* is eliminated in the reduction $p\\totom q$: then the of *c* in *q* are the cuts $\\cut{t\\_i}{s\\_i}\\in\\cuts(q)$ for 1 ≤ *i* ≤ *n*. For any edge *e*, we write (*e*) for any length 1 path *t* ∼ *e**s*. If $\\chi\\in \\paths(q)$, a of *χ* is any subpath (*d*)*ξ*(*d*ʹ) where *d* and *d*ʹ are (necessarily distinct) residuals of the same cut in *p*. In the remaining of this section, we show that a path in *q* is necessarily obtained by alternating paths (essentially) in *p* and slipknots in *q*, that recursively consist of such alternations. This will allow us to bound ln(*q*) depending on ln(*p*), by reasoning inductively on these paths.\nPreserved paths\n---------------\nNotice that $\\ST(q)\\subseteq\\ST(p)$ and, given a switching *J* of *q*, it is always possible to extend *J* into a switching *I* of *p*: to determine *I* uniquely amounts to select a premise for each $\\parr$-tree in an eliminated cut.\nLet *J* be a switching of *q* and *I* an extension of *J* on *p*. Observe that if *t* ∼ *e**p*, *I**t*ʹ and neither *t* nor *t*ʹ is an element of an eliminated cut, then *e* is also an edge of *q* that is not a residual cut; conversely, if *t* ∼ *e**q*, *J**t*ʹ and *e* is not a residual cut, then *e* is also an edge of *p*. We then say the edge *e* is by the reduction $p\\totom q$. If a preserved edge *e* is a cut, an axiom, a ⊗ -edge or a $\\parr$-edge, then *e* has the same endpoints in *p* and in *q*: *t* ∼ *e**p*, *I**t*ʹ iff *t* ∼ *e**q*, *J**t*ʹ. If $e=\\w$ is a jump, one endpoint might be changed: indeed, $\\w\\sim^p\\_e \\jump\\_p(\\w)$ and $\\w\\sim^q\\_e \\jump\\_q(\\w)$, and we might have $\\jump\\_p(\\w)\\not=\\jump\\_q(\\w)$ when $\\jump\\_p(\\w)$ is part of an eliminated cut. In this case, we say *e* is a. We say $u\\in\\ST(p)$ is an of $v\\in\\ST(q)$, if either *u* = *v* or *u* is involved in an eliminated cut $c=\\cut{u}{u'}$, and either $u=\\otimes(\\vec t)$ and $v\\in\\vec t$ or $u=\\parr(\\vec s)$ and $v\\in\\vec s$.\n[lem:nearcut] Assume $(d)\\chi(d')\\in\\paths(q)$ and *d* and *d*ʹ are residuals. Then *χ* is non empty, and its first and last edges are preserved.\nThis is a direct consequence of the fact that if (*c*)(*e*) is a path and *c* is a cut then *e* is not a cut.\nObserve that even if a path $\\chi\\in\\paths(q)$ crosses preserved edges only, it is not sufficient to have $\\chi\\in\\paths(p)$, because the endpoints of redirected jumps might change. We say *χ* is a if *χ* crosses preserved edges only, and we can write either *χ* = *χ*ʹ or $\\chi=(t\\sim\\_{\\w}\\counit{\\w})\\chi'$ or $\\chi'(\\counit{\\w'}\\sim\\_{\\w'} t')$ or $\\chi=(t\\sim\\_{\\w}\\counit{\\w})\\chi'(\\counit{\\w'}\\sim\\_{\\w'} t')$ where *χ*ʹ crosses no redirected jump.\n[lem:preserved] Any non empty preserved path $\\chi:t\\leadsto\\_q t'$ induces a unique preserved path $\\chi^-:s\\leadsto\\_p s'$ with the same sequence of edges: in particular, $\\chi^-\\in\\paths(p,I)$ as soon as $\\chi\\in\\paths(q,J)$ and *I* is an extension of *J*; and *s* (resp. *s*ʹ) is an anchor of *t* (resp. *t*ʹ’). Moreover, if *χ*1*χ*2 is a preserved path then $(\\chi\\_1\\chi\\_2)^-=\\chi^-\\_1\\chi^-\\_2\\in\\paths(p)$.\nThe first part is a direct consequence of the definition. If moreover *χ*1*χ*2 is a preserved path, and neither *χ*1 nor *χ*2 is an empty path, then neither the last edge of *χ*1 nor the first edge of *χ*2 can be a redirected jump.\nBy convention, if *χ* is empty, we set *χ*− = *χ*. We say *χ*− is the path of *p* *χ*. In the next two subsections, we extend the generation of paths in *p* from paths in *q*, first to paths without slipknots, then to arbitrary paths.\nBridges and straight paths\n--------------------------\nLet $c=\\cut{t\\_0}{s\\_0}\\in \\cuts(p)$ with $t\\_0=\\otimes(\\vec t)$ and $s\\_0=\\parr(\\vec s)$. We say an edge *e* is if either *e* = *c*, or *e* = (*t*0, *t*) with $t\\in\\vec t$, or *e* = *s*0. And we say $\\chi\\in\\paths(p)$ is bound to *c* if all the edges crossed by *χ* are bound to *c*. Observe that $e\\in\\paths(p)$ is a preserved edge iff it is not bound to an eliminated cut.\n[lem:bound] Let $c=\\cut{t\\_0}{s\\_0}$:\n* if $\\chi\\in\\paths(p)$ is bound to $c=\\cut{t\\_0}{s\\_0}$ and $s\\_0=\\parr(\\vec s)$ then there exists $s\\in\\vec s$ such that $\\chi\\in\\paths(p,I)$ whenever *I*(*s*0) = *s*;\n* if $\\chi\\in\\paths(p,I)$ does not cross any edge bound to *c* then $\\chi\\in\\paths(p,I')$ whenever *I* and *I*ʹ differ only on *s*0.\nIt is sufficient to observe that the edge *s*0 is bound to *c*; and the only edge $e\\in\\parrs(p)$ that may be visited by a path bound to *c* is *s*0.\nA is a path *χ* that is bound to *c* and that crosses *c*. Observe that *χ* is a *c*-bridge iff either *χ* or the reverse path $\\reverse \\chi$ is a subpath of some *t* ∼ (*t*0, *t*)*t*0 ∼ *c**s*0 ∼ *s*0*s* with $t\\in\\vec\nt$ and $s\\in\\vec s$. Moreover, given $t\\in \\{ t\\_0\\}\\cup \\vec t$ and $s\\in \\{s\\_0\\}\\cup \\vec s$ there is a unique *c*-bridge $t\\leadsto s$.\n[lem:bridge] Assume $\\chi\\_1\\xi\\chi\\_2\\in\\paths(q)$ and $\\xi=t\\sim\\_{\\cut ts} s$, where *χ*1 and *χ*2 are preserved paths, $t\\in\\vec t$ and $s\\in\\vec s$. Then there exists a *c*-bridge $\\bridge\\xi$ such that $\\chi\\_1^-\\bridge\\xi\\chi\\_2^-\\in\\paths(p)$.\nWrite $\\chi\\_1:v\\_1\\leadsto\\_{q,J} t$ and $\\chi\\_2:s\\leadsto\\_{q,J} v\\_2$: by Lemma [lem:preserved], we obtain $\\chi\\_1^-:u\\_1\\leadsto\\_{p,I} t'$ and $\\chi\\_2^-:s'\\leadsto\\_{p,I} u\\_2$ where *u*1, *t*ʹ, *s*ʹ and *u*2 are anchors of *v*1, *t*, *s* and *v*2 respectively, for any extension *I* of *J*. In particular, *t*ʹ ∈ {*t*, *t*0} and *s*ʹ ∈ {*s*, *s*0} and we can fix $\\bridge\\xi:t'\\leadsto s'$ to be the only *c*-bridge with those endpoints. Observe indeed that $\\bridge\\xi\\in\\paths(p,I)$ as soon as *I*(*s*0) = *s*. Then by Lemmas [lem:preserved] and [lem:bound], we can concatenate $\\chi\\_1^-\\bridge\\xi\\chi\\_2^-:u\\_1\\leadsto\\_{p,I} u\\_2$.\nDespite the notation, the definition of $\\bridge\\xi$ does depend on *χ*1 and *χ*2: whenever we use Lemma [lem:bridge], however, the values of *χ*1 and *χ*2 should be clear from the context.\nWe say a path $\\chi\\in\\paths(q)$ is a if it has no slipknot. Such a path is essentially a path of *p*, up to replacing residuals with bridges:\n[lem:straight] If *χ* is a straight path, there exists a unique sequence of pairwise distinct eliminated cuts $c\\_1,\\dotsc,c\\_n\\in\\cuts(p)\\setminus\\cuts(q)$, such that we can write *χ* = *χ*1(*d*1)⋯*χ**n*(*d**n*)*χ**n* + 1 where each *d**i* is a residual of *c**i* and *χ* crosses no other residual. Moreover we can form $\\chi^-=\\chi^-\\_1\\bridge{(d\\_1)}\\cdots\\chi^-\\_n\\bridge{(d\\_n)}\\chi\\_{n+1}^-\\in\\paths(p)$.\nThe first part is straightforward reformulation of the absence of slipknots. The second part follows by applying Lemma [lem:bridge] to each *χ**i*(*d**i*)*χ**i* + 1 (or the reverse path): the concatenation of preserved paths and bridges in the definition of *χ*− is allowed by Lemmas [lem:preserved] and [lem:bound].\nWe say two paths $\\chi\\_1,\\chi\\_2\\in\\paths(q)$ are if they are disjoint and there is no eliminated cut *c* such that both *χ*1 and *χ*2 cross a residual of *c*.\n[lem:compatstraight] Assume $\\chi\\_1,\\dotsc,\\chi\\_n\\in\\paths(q,J)$ are pairwise independent straight paths and $c\\_1,\\dotsc,c\\_k\\in\\cuts(p)\\setminus\\cuts(q)$ are such that no *χ**i* crosses a residual of *c**j*, for 1 ≤ *i* ≤ *n* and 1 ≤ *j* ≤ *k*. Then for any $\\xi\\_1,\\dotsc,\\xi\\_k\\in\\paths(p)$ such that *ξ**i* is bound to *c* for 1 ≤ *i* ≤ *k*, there exists an extension *I* of *J* such that $\\chi\\_i^-\\in\\paths(p,I)$ for 1 ≤ *i* ≤ *n* and $\\xi\\_j\\in\\paths(p,I)$ for 1 ≤ *j* ≤ *k*.\nIt is sufficient to observe that if *χ*− crosses an edge bound to an eliminated cut *c* then *χ* crosses a residual of *c*. Then the result is a direct consequence of the definition of *χ**i*−, together with Lemmas [lem:preserved] and [lem:bound].\nThe generation of a path is thus compatible with the concatenation of independent straight paths:\n[lem:concatstraight] If $\\chi\\_1:v\\leadsto\\_q u$ and $\\chi\\_2:u\\leadsto\\_q t$ are independent straight paths then *χ*1*χ*2 is a straight path and (*χ*1*χ*2)− = *χ*1−*χ*2−.\nThat *χ*1*χ*2 is a straight path follows directly from the hypotheses. Write *χ**i* = *χ*1*i*(*d*1*i*)⋯*χ**n**i**i*(*d**n**i**i*)*χ**n**i* + 1*i*: it is then sufficient to apply the definition of *χ**i*− and observe that (*χ**n*1 + 11*χ*12)− = (*χ**n*1 + 11)−(*χ*12)− by Lemma [lem:preserved].\nBounces and slipknots\n---------------------\nLet $c=\\cut{t\\_0}{s\\_0}\\in \\cuts(p)$ with $t\\_0=\\otimes(\\vec t)$. A is a path *χ* that is bound to *c*, that does not cross *c* and that visits *t*0: *χ* is either the empty path *ε**t*0, or *t*0 ∼ *t*0, *t**t* or *t* ∼ *t*0, *t**t*0 with $t\\in\\vec t$, or *t* ∼ (*t*0, *t*)*t*0 ∼ (*t*0, *t*ʹ)*t*ʹ with $t\\not=t'\\in\\vec t$. Given $t,t'\\in \\{t\\_0\\}\\cup\\vec t$, such that either *t* = *t*0 or *t*ʹ = *t*0 or $t\\not= t'$, there is a unique *c*-bounce $t\\leadsto t'$.\n[lem:bounce] Assume $\\chi\\_1\\xi\\chi\\_2\\in\\paths(q)$ and $\\xi=(t\\_1\\sim\\_{\\cut{t\\_1}{s\\_1}} s\\_1)\\xi'(s\\_2\\sim\\_{\\cut{t\\_2}{s\\_2}} t\\_2)$, where *χ*1 and *χ*2 are preserved paths, $t\\_1,t\\_2\\in\\vec t$ and $s\\_1,s\\_2\\in\\vec s$. Then there exists a *c*-bounce $\\bounce\\xi$ such that $\\chi\\_1^-\\bounce\\xi\\chi\\_2^-\\in\\paths(p)$.\nNecessarily, $\\cut{t\\_1}{s\\_1}\\not=\\cut{t\\_2}{s\\_2}$, hence $t\\_1\\not=t\\_2$. Write $\\chi\\_1:v\\_1\\leadsto\\_{q} t\\_1$ and $\\chi\\_2:t\\_2\\leadsto\\_{q} v\\_2$: we obtain $\\chi\\_1^-:u\\_1\\leadsto\\_{p} t'\\_1$ and $\\chi\\_2^-:t'\\_2\\leadsto\\_{p} u\\_2$ where *u*1, *t*ʹ1, *t*ʹ2 and *u*2 are anchors of *v*1, *t*1, *t*2 and *v*2 respectively, for any extension *I* of *J*. In particular, *t*ʹ1 ∈ {*t*1, *t*0} and *t*ʹ2 ∈ {*t*2, *t*0} with $t\\_1\\not=t\\_2$, and we can fix $\\bounce\\xi:t'\\_1\\leadsto t'\\_2$ to be the only the only *c*-bounce with those endpoints. Then we can concatenate $\\chi\\_1^-\\bounce\\xi\\chi\\_2^-:u\\_1\\leadsto\\_p u\\_2$ by Lemmas [lem:preserved] and [lem:bound].\nAgain, the definition of $\\bounce\\xi$ does depend on *χ*1 and *χ*2 but these should be clear from the context when we use Lemma [lem:bounce].\nWe are now ready to prove that paths in *q* are alternations of straight paths and slipknots, and generate paths in *p* by replacing slipknots with bounces:\n[thm:paths] For each path $\\chi\\in\\paths(q)$, there exists a unique sequence of pairwise distinct eliminated cuts $c\\_1,\\dotsc,c\\_n\\in\\cuts(p)\\setminus\\cuts(q)$, such than we can write *χ* = *χ*1*ξ*1⋯*χ**n**ξ**n**χ**n* + 1 where:\n* each *χ**i* is a straight path that crosses no residual of *c**j* for 1 ≤ *j* ≤ *n*;\n* *χ**i* and *χ**j* are independent when $i\\not=j$;\n* each *ξ**i* is a slipknot $(d\\_i)\\xi'\\_i(d'\\_i):t\\_i\\leadsto t'\\_i$ where *d**i* and *d*ʹ*i* are residuals of *c**i*, and *t**i* and *t*ʹ*i* are distinct premises of the ⊗ -tree of *c**i*.\nMoreover $\\chi^-=\\chi^-\\_1\\bounce{\\xi\\_1}\\cdots\\chi^-\\_n\\bounce{\\xi}\\_n\\chi^-\\_{n+1}\\in\\paths(p)$.\n[ baseline=(cdots.base), ]\n(tensbas) at (0,0) ⊗ ; (parbas) at (1,0)$\\parr$; at(0.5,-0.8)*c**n*;\n(tenshaut) at (1,4) ⊗ ; (parhaut) at (2,4)$\\parr$; at(1.5,3.2)*c*1; (-1,0.5)–(-0.6,0.5); (-0.5,0.5)–(-0.3,0.5)–(tensbas)–(0,-0.5)–(1,-0.5)–(parbas)–(1,1)–(2,1)–(2,0.8); (parbas)–(1.5,0)–(1.5,0.3)–(1.8,0.3)–(1.8,-0.2)–(2,-0.2); (2,0)–(2,0.6); (tensbas)–(0.5,0.3)–(0.5,0.5); (0.5,0.5)–(0.5,4); (0.5,4.3)–(0.5,4.5)–(0.8,4.5)–(tenshaut); (tenshaut)–(1,3.5)–(2,3.5)–(parhaut)–(2,5)–(3,5)–(3,4.8); (parhaut)–(2.5,4)–(2.5,4.3)–(2.8,4.3)–(2.8,3.8)–(3,3.8); (3,4)–(3,4.6); (tenshaut)–(1.3,4.3)–(1.3,4.5); (1.3,4.7)–(1.3,5); at(1.1,5.2)*χ*1−;\nat(-0.8,0.8)*χ**n* + 1−;\nreduces to\n[baseline=(cdots.base)]\n(-1,0.5)–(-0.6,0.5); (-0.5,0.5)–(-0.3,0.5)–(-0.3,-0.5)–(1.5,-0.5)–(1.5,0)–(1.5,0.3)–(1.8,0.3)–(1.8,-0.2)–(2,-0.2); (0.5,0.5)–(0.5,0)–(1,0)–(1,1)–(2,1)–(2,0.8); (2,0)–(2,0.6); (0.5,0.5)–(0.5,4); (3,4)–(3,4.6); (0.5,4.3)–(0.5,4.5)–(1,4.5)–(1,3.5)–(2.5,3.5)–(2.5,4)–(2.5,4.3)–(2.8,4.3)–(2.8,3.8)–(3,3.8); (1.5,4.5)–(1.5,4)–(2,4)–(2,5)–(3,5)–(3,4.8); (1.5,4.7)–(1.5,5); at(1.8,4.3)*d*1; at(1.8,3.2)*d*1ʹ;\nat(3.3,4.3)*ξ*1ʹ; at(1.2,5.2)*χ*1;\nat(0.8,0.3)*d**n*; at(0.8,-0.8)*d**n*ʹ;\nat(2.3,0.3)*ξ**n*ʹ; at(-0.8,0.8)*χ**n* + 1;\n[fig:pathsslipknots]\nFigure [fig:pathsslipknots] illustrates the relationship between *χ*− and *χ*, in the simple case where each cut is between binary connectives, and no redirected jump is involved: each *χ**i*− bounces on the ⊗ side of *c**i* and joins *χ**i* + 1− directly instead of crossing the cut.\nThe proof of Theorem [thm:paths] is by induction on the length of *χ*. We break it down into a series of intermediate results. Formally, given $\\chi\\in\\paths(q)$:\n* we first establish Lemma [lem:compat] for all paths such that Theorem [thm:paths] holds;\n* we deduce Lemma [lem:loop] for *χ* from Theorem [thm:paths] and Lemma [lem:compat] applied to *χ*;\n* Lemma [lem:slipknot] for *χ* is a direct consequence of Lemma [lem:loop] applied to strict subpaths of *χ*;\n* we prove Lemma [lem:bracketing] for *χ* by applying Theorem [thm:paths], Lemma [lem:compat] and Lemma [lem:slipknot] to strict subpaths of *χ*;\n* then we prove Theorem [thm:paths] for *χ* by applying Lemmas [lem:slipknot] and [lem:bracketing] to *χ*, and Lemma [lem:compat] to strict subpaths of *χ*.\n[lem:compat] Assume $\\chi\\_1,\\dotsc,\\chi\\_n\\in\\paths(q,J)$ are pairwise disjoint paths, and $c\\_1,\\dotsc,c\\_k\\in\\cuts(p)\\setminus\\cuts(q)$, are such that no *χ**i* crosses a residual of *c**j*, for 1 ≤ *i* ≤ *n* and 1 ≤ *j* ≤ *k*. Then for any $\\xi\\_1,\\dotsc,\\xi\\_k\\in\\paths(p)$ such that *ξ**i* is bound to *c* for 1 ≤ *i* ≤ *k*, there exists an extension *I* of *J* such that $\\chi\\_i^-\\in\\paths(p,I)$ for 1 ≤ *i* ≤ *n* and $\\xi\\_j\\in\\paths(p,I)$ for 1 ≤ *j* ≤ *k*.\nThis is a direct consequence of the definition of *χ**i*− in Theorem [thm:paths], together with Lemma [lem:compatstraight].\n[lem:loop] Assume $d=\\cut{t}{s}$ and $d'=\\cut{t'}{s'}$ are distinct residuals of the same cut $c=\\cut{t\\_0}{s\\_0}$ with $t\\_0=\\otimes(\\vec t)$, $s\\_0=\\parr(\\vec s)$, $t,t'\\in\\vec t$ and $s,s'\\in\\vec s$. If *u* ∈ *d*, *u*ʹ ∈ *d*ʹ and $\\chi:u\\leadsto\\_q u'$ crosses no residual of *c* then *u* = *s* and *u*ʹ = *s*ʹ.\nWrite *χ* = *χ*1*ξ*1⋯*χ**n**ξ**n**χ**n* + 1 as in Theorem [thm:paths], and $\\chi^-:v\\leadsto\\_{p,I} v'$ where *v* (resp. *v*ʹ) is an anchor of *u* (resp. *u*ʹ). Observe that *χ* is non empty because $u\\not=u'$. Moreover, the first edge of *χ* is a preserved edge: if *χ* = (*u* ∼ *e**v*)*χ*ʹ then *e* cannot be a cut, as otherwise we would have *e* = *d* and *χ* would cross a residual of *c*. It follows that *χ*1− is non empty, hence *χ*− is non empty. Since *χ*− is non empty, there is no path $\\zeta:v\\leadsto v'$ bound to *c*: otherwise, by Lemma [lem:compat], we could form a non empty cycle $\\reverse\\zeta\\chi^-:v'\\leadsto\\_p v'$.\nIf *u* = *t* and *u*ʹ = *t*ʹ, we have *v* ∈ {*t*, *t*0} and *v*ʹ ∈ {*t*ʹ, *t*0} with $t\\not=t'$, and we obtain a *c*-bounce $v\\leadsto v'$, hence a contradiction. If *u* = *t* and *u*ʹ = *s*ʹ, we have *v* ∈ {*t*, *t*0} and *v*ʹ ∈ {*s*ʹ, *s*0}, and we obtain a *c*-bridge $v\\leadsto v'$, hence a contradiction. We rule out the case *u* = *s* and *u*ʹ = *t*ʹ symmetrically.\n[lem:slipknot] If $\\chi\\in\\paths(q)$ and $c=\\cut{\\otimes(\\vec t)}{\\parr(\\vec s)}\\in\\cuts(p)\\setminus\\cuts(q)$, then *χ* crosses at most two residuals of *c*, and in this case we can write $\\chi=\\chi\\_1(t\\sim\\_{\\cut ts} s)\\chi\\_2(s'\\sim\\_{\\cut{t'}{s'}}t')\\chi\\_3$ with $t,t'\\in\\vec t$ and $s,s'\\in\\vec s$.\nIf *χ* = *χ*1(*d*)*χ*2(*d*ʹ)*χ*3, where $d=\\cut{t}{s}$ and $d'=\\cut{t'}{s'}$ are residuals of *c* with $t,t'\\in\\vec t$ and $s,s'\\in\\vec s$, and if *χ*2 crosses no residual of *c*, then by Lemma [lem:loop] applied to *χ*2, we obtain $\\chi=\\chi\\_1(t\\sim\\_{\\cut ts} s)\\chi\\_2(s'\\sim\\_{\\cut{t'}{s'}}t')\\chi\\_3$. If moreover *χ*1 (resp. *χ*3) crossed another residual of *c*, we would obtain a contradiction by applying Lemma [lem:loop] to a strict subpath of *χ*1 (resp. *χ*3) hence of *χ*.\n[lem:bracketing] Slipknots are well-bracketed in the following sense: there is no path $\\chi=(d\\_1)\\chi\\_1(d\\_2)\\chi\\_2(d'\\_1)\\chi\\_3(d'\\_2)\\in\\paths(q)$ such that, for 1 ≤ *i* ≤ 2, *d**i* and *d*ʹ*i* are residuals of the same cut.\nAssume $\\chi=(d\\_1)\\chi\\_1(d\\_2)\\chi\\_2(d'\\_1)\\chi\\_3(d'\\_2)\\in\\paths(q)$ such that, for 1 ≤ *i* ≤ 2, *d**i* and *d*ʹ*i* are residuals of the same cut. We can assume w.l.o.g. that *χ*1 and *χ*3 are independent: otherwise there is a prefix of *χ* with this additional property.\nFor 1 ≤ *i* ≤ 2, let $c\\_i=\\cut{t\\_0^i}{s\\_0^i}$, with $t^i\\_0=\\otimes(\\vec t\\_i)$ and $s^i\\_0=\\parr(\\vec s\\_i)$, and assume *d**i* = {*t**i*, *s**i*} and *d*ʹ*i* = {*t*ʹ*i*, *s*ʹ*i*}, with $t\\_i,t'\\_i\\in\\vec t\\_i$ and $s\\_i,s'\\_i\\in\\vec s\\_i$. By Lemma [lem:slipknot], we must have *χ* = (*t*1 ∼ *d*1*s*1)*χ*1(*t*2 ∼ *d*2*s*2)*χ*2(*s*ʹ1 ∼ *d*ʹ1*t*ʹ1)*χ*3(*s*ʹ2 ∼ *d*ʹ2*t*ʹ2).\nBy Theorem [thm:paths], we obtain $\\chi\\_1^-:u\\_1\\leadsto\\_p v\\_2$ and $\\chi\\_3^-:v\\_1\\leadsto\\_p u\\_2$, where *u*1 ∈ {*s*1, *s*01}, *v*2 ∈ {*t*2, *t*02}, *v*1 ∈ {*t*ʹ1, *t*01} and *u*2 ∈ {*s*ʹ2, *s*02}. Write $\\zeta\\_1:v\\_1\\leadsto\\_p u\\_1$ (resp. $\\zeta\\_2:v\\_2\\leadsto\\_p u\\_2$) for the only *c*1-bridge (resp. *c*2-bridge) with those endpoints. Then we obtain a non empty cycle $\\chi^-\\_1\\zeta\\_2(\\reverse{\\chi\\_3})^-\\zeta\\_1:u\\_1\\leadsto\\_p u\\_1$: the concatenation is allowed by Lemma [lem:compat] applied to *χ*1 and $\\reverse{\\chi\\_3}$.\n[Proof of Theorem [thm:paths]] By Lemma [lem:bracketing], we can write *χ* = *χ*1*ξ*1⋯*χ**n**ξ**n**χ**n* + 1 where $\\xi\\_1,\\dotsc,\\xi\\_n$ are the slipknots of *χ* that are maximal (*i.e.* not strict subpaths of other slipknots) and this writing is unique. Let $c\\_1,\\dotsc,c\\_n$ be the associated eliminated cuts. By Lemma [lem:slipknot], the *c**j*’s are pairwise distinct, and $\\xi\\_j:t\\_j\\leadsto\\_p t'\\_j$ where *t**j* and *t*ʹ*j* are distinct premises of the ⊗ -tree of *c**j*. Moreover, since each slipknot of *χ* is a subpath of some *ξ**j*:\n* each *χ**i* is a straight path and it crosses no residual of any *c**j*;\n* the *χ**i*’s are pairwise independent.\nIf *n* = 0, we can set *χ*− = *χ*1−. Otherwise, we apply Lemma [lem:compat] to the *χ**i*’s, which are strict subpaths of *χ*, which allows to concatenate $\\chi^-=\\chi^-\\_1\\bounce{\\xi\\_1}\\cdots\\chi^-\\_n\\bounce{\\xi}\\_n\\chi^-\\_{n+1}\\in\\paths(p)$.\nAgain, the construction of *χ*− is compatible with the concatenation of independent paths:\n[lem:concat] If $\\chi\\_1\\chi\\_2\\in\\paths(q)$ and *χ*1 and *χ*2 are independent, then (*χ*1*χ*2)− = *χ*1−*χ*2−.\nAs for Lemma [lem:concatstraight], this is a direct consequence of the definition of (*χ*1*χ*2)−, *χ*1− and *χ*2−, this time using Lemma [lem:concatstraight] to concatenate a straight suffix of *χ*1 and a straight prefix of *χ*2.\nBounding the growth of ln\n-------------------------\nNow we show that we can bound ln(*q*) depending only on ln(*p*). We first need some basic properties relating the length of *χ*− with that of *χ*.\n[lem:length] Let $\\chi,\\xi\\in\\paths(q)$ and $\\zeta\\in\\paths(p)$:\n1. if *χ* is preserved then ln(*χ*−) = ln(*χ*);\n2. if *ζ* is a bridge then 1 ≤ ln(*ζ*) ≤ 3;\n3. if *ζ* is a bounce then ln(*ζ*) ≤ 2;\n4. if *χ* is straight, then ln(*χ*) ≤ ln(*χ*−) ≤ 3ln(*χ*);\n5. if *ξ* is a prefix of *χ* and *χ* is straight then *ξ*− is a prefix of *χ*−;\n6. if *ξ* is a slipknot then ln(*ξ*) ≥ 3 and $\\ln(\\bounce\\xi)<\\ln(\\xi)$;\n7. in general ln(*χ*−) ≤ 3ln(*χ*).\nThe first three properties are direct consequences of the definitions. Item (4) follows from (1) and (2). Item (5) follows from Lemma [lem:concatstraight]. Item (6) follows from Lemma [lem:nearcut] and (3). And item (7) follows from (4) and (6).\nObserve that in general, we do not have ln(*ζ*−) ≤ ln(*χ*−) when *ζ* is a prefix of *χ*: *ζ* may enter an arbitrarily long slipknot of *χ* that is replaced by a single bounce in *χ*−. For this reason, we introduce the following notion: if $\\chi\\in \\paths(q)$, we define the of *χ* (relative to the reduction $p\\totom q$ we consider) by $\\width(\\chi)=\\max\\{\\ln(\\zeta^-)\\tq \\zeta \\text{ prefix of } \\chi\\}$.\n[lem:width] For any path $\\chi\\in\\paths(q)$, $\\ln(\\chi^-)\\le \\width(\\chi)\\le\\ln(p)$ and $\\width(\\chi)\\le 3\\ln(\\chi)$. Moreover, if *ζ* is a prefix of *χ*, we have $\\width(\\zeta)\\le\\width(\\chi)$. If moreover *χ* is straight, $\\width(\\chi)=\\ln(\\chi^-)\\ge\\ln(\\chi)$.\nWe obtain $\\ln(\\chi^-)\\le \\width(\\chi)\\le\\ln(p)$ and $\\width(\\zeta)\\le\\width(\\chi)$ directly from the definition of width. Item (7) of Lemma [lem:length] gives $\\width(\\chi)\\le 3\\ln(\\chi)$. If *χ* is straight $\\width(\\chi)=\\ln(\\chi^-)\\ge\\ln(\\chi)$ follows from items (4) and (5) of Lemma [lem:length].\nDefine $\\varphi:\\N\\to\\N$ inductively by *φ*(0) = 0 and *φ*(*n*) = *n* + (*n* + 1)(*φ*(*n* − 1) + 2) if *n* > 0. Observe that *n* ≤ *φ*(*n*) ≤ *φ*(*n* + 1).\n[lem:generatedlength] If $\\chi\\in\\paths(q)$ then $\\ln(\\chi)\\le\\varphi(\\width(\\chi))$.\nThe proof is by induction on $\\width(\\chi)$. If *χ* is straight then, by Lemma [lem:width], $\\ln(\\chi)\\leq\\width(\\chi)\\le\\varphi(\\width(\\chi))$.\nWrite *χ* = *χ*1*ξ*1⋯*χ**n**ξ**n**χ**n* + 1 as in Theorem [thm:paths]: we have ln(*χ*) = ∑*i* = 1*n* + 1ln(*χ**i*) + ∑*j* = 1*n*ln(*ξ**j*). Since $\\chi^-=\\chi\\_1^-\\bounce{\\xi\\_1}\\cdots\\chi\\_n^-\\bounce{\\xi\\_n}\\chi\\_{n+1}^-$, we have ∑*i* = 1*n* + 1ln(*χ**i*−) ≤ ln(*χ*−). Since each *χ**i* is straight, we obtain ln(*χ*) ≤ ln(*χ*−) + ∑*j* = 1*n*ln(*ξ**j*) from the previous inequality, by applying item (4) of Lemma [lem:length].\nMoreover observe that, by Lemma [lem:nearcut], *χ**i* is non empty for 1 < *i* < *n* + 1. Hence ln(*χ*−) ≥ *n* − 1, and we obtain $n\\le\\width(\\chi)+1$.\nIt remains to bound ln(*ξ**j*) for 1 ≤ *j* ≤ *n*. We can write *ξ**j* = (*d**j*)*χ*ʹ*j*(*d*ʹ*j*) where *d**j* and *d*ʹ*j* are the residuals of the cut *c**j* associated with *ξ**j*. Let *ζ*ʹ*j* be a prefix of *χ*ʹ*j* and write *ζ**j* = *χ*1*ξ*1⋯*χ**j*(*d**j*)*ζ*ʹ*j* which is a prefix of *χ*. Observe that, by Theorem [thm:paths], *c**j* has no residual in *ζ**j* other than *d**j*, and *χ*1*ξ*1⋯*χ**j*(*d**j*) and *ζ*ʹ*j* are independent. Hence *χ**j*(*d**j*) is straight and $\\zeta\\_j^-=\\chi\\_1^-\\bounce{\\xi\\_1}\\cdots\\chi\\_j^-\\bridge{d\\_j}(\\zeta'\\_j)^-$ follows by Lemma [lem:concat]. Since $\\ln(\\bridge{d\\_j})\\ge 1$, we obtain $\\ln((\\zeta'\\_j)^-)\\le \\ln(\\zeta\\_j^-)-1\\le\\width(\\chi)-1$.\nHence $\\width(\\chi'\\_j)\\le\\width(\\chi)-1$: we apply the induction hypothesis and obtain $\\ln(\\chi'\\_j)\\le\\varphi(\\width(\\chi'\\_j))\\le\\varphi(\\width(\\chi)-1)$ because *φ* is monotonous. It follows that $\\ln(\\xi\\_j)\\le \\varphi(\\width(\\chi)-1)+2$, and we conclude:\n \n$$\\begin{aligned}\n \\ln(\\chi)\n &\\le\\ln(\\chi^-)+\\sum\\_{j=1}^n\\ln(\\xi\\_j)\n \\\\\n &\\le\\width(\\chi)+(\\width(\\chi)+1)(\\varphi(\\width(\\chi)-1)+2)\n \\\\\n &\\le\\varphi(\\width(\\chi)).\n \\qedhere\n \\end{aligned}$$ \nUsing Lemma [lem:width] again, we obtain:\n[vc] Let $p\\totom q$. Then, ln(*q*) ≤ *φ*(ln(*p*)).\nNotice that the previous result can be seen as a quantitative version of the preservation of acyclicity in proof nets under reduction. In the following example, we illustrate how acyclicity is mandatory for the existence of a function *φ* as in Corollary [vc].\n[ex:cycle1] Let $p=(\\cut{t}{s};)$ with $t=\\otimes(x\\_0,\\dotsc,x\\_n)$ and $s=\\parr(\\bar x\\_1,\\dotsc,\\bar x\\_n,\\bar x\\_0)$: by setting $I(s)=\\bar x\\_i$, we obtain a cycle $x\\_i\\sim\\_{\\{x\\_i,\\bar x\\_i\\}}\\bar x\\_i\\sim\\_s s\\sim\\_{\\cut{t}{s}} t\\sim\\_{t,x\\_i} x\\_i$. Since each path $\\chi\\in\\paths(p)$ can cross each of *s* and $\\cut{t}{s}$ at most once, it is easy to check that ln(*p*) = 6.\nBut $p\\totom q=(\\cut{x\\_0}{\\bar x\\_1},\\dotsc,\\cut{x\\_{n-1}}{\\bar x\\_n},\\cut{x\\_{n}}{\\bar x\\_0};)$ hence ln(*q*) = 2(*n* + 1). The situation is illustrated in Figure [fig:cycle1].\n(0)at(0,0)ax; (1)at((0) + (0, − 1))ax; (2)at((1) + (0, − .7))$\\rvdots$; (3)at((2) + (0, − .7))ax; (t)at((3) + ( − 1, − 1.5)) ⊗ ; (p)at((3) + (1, − 1.5))$\\parr$; (c)at((0, − 1) + 1/2 \\* ((t)+(p)))cut;\n(0)node*x*0 ; (0)node$\\ \\bar x\\_0$; (1)node*x*1 ; (1)node$\\ \\bar x\\_1$; (3)node*x**n* ; (3)node$\\ \\bar x\\_n$;\n(t)to[out=120,in=west](0); (t)to[out=100,in=west](1); (t) to [out=80,in=west] node [sloped, near start] $\\rvdots$ (2); (t)to[out=60,in=west](3); (p)to[out=125,in=east](1); (p) to [out=95,in=east] node [sloped, very near start] $\\rvdots$ (2); (p)to[out=60,in=east](3); (p)to[out=50,in=east](0); (t)to[out=south,in=west](c); (p)to[out=south,in=east](c);\n(0)at(6,0)ax; (1)at((0) + (0, − 1))ax; (2)at((1) + (0, − 1))$\\rvdots$; (3)at((2) + (0, − 1))ax;\n(c01)at((3) + (0, − 1.2))cut; (c12)at((*c*01) + (0, − 1))cut; (c23)at((*c*12) + (0, − 1))$\\rvdots$; (c30)at((*c*23) + (0, − 1))cut;\n(0)node*x*0 ; (0)node$\\ \\bar x\\_0$; (1)node*x*1 ; (1)node$\\ \\bar x\\_1$; (3)node*x**n* ; (3)node$\\ \\bar x\\_n$;\n(0)to[out=west,in=west](c01); (1)to[out=east,in=east](c01); (1)to[out=west,in=west](c12); (c12) to [out=east,in=east] node [very near end,inner sep=0pt,sloped] (c12a) $\\scriptstyle\\cdots$ node [very near end,inner sep=0pt] (c12b) (2); (c12) to [out=east,in=-40] (c12a.east) ; (3) to [out=east,in=east] node [very near end,inner sep=0pt,sloped] (c23a) $\\scriptstyle\\cdots$ node [very near end,inner sep=0pt] (c23b) (c23); (3) to [out=east,in=40] (c23a.east) ; (3)to[out=west,in=west](c30); (0)to[out=east,in=east](c30);\nat (3,-3.5)$\\totom$;\n[fig:cycle1]\nErratum\n-------\nIn the extended abstract of the present paper, the analogue of Subsection [subsection:ln] claimed to establish similar results for another measure on paths: rather than the length ln(*ξ*) of a path *ξ*, we considered the number $\\vc(\\xi)$ of all the cuts $\\cut{t}{s}$ such that *ξ* visits *t* or *s*.\nIt is easy to check that Lemma [ax] still holds if we replace the maximal length of a path with the maximum number of cuts in a chain of axiom cuts in the sense of Figure [fig:collapse]. Given the situation depicted in Figure [fig:pathsslipknots], however, it is evident that a bound on the number of cuts crossed by a path cannot be preserved: the path on the left hand side crosses no cut, while the path in the reduct crosses an arbitrary number of (possibly axiom) cuts. We introduced $\\vc(\\xi)$ in our previous attempt, precisely to capture this example: if *ξ* is the path following the tensors of the left hand side, then $\\vc(\\xi)\\ge n$. But this fix is actually not sufficient: if we replace each $c\\_i=\\cut{t\\_i}{s\\_i}$ in Figure [fig:pathsslipknots], with $\\cut{\\otimes(t\\_i,s\\_i)}{\\parr(x\\_i,\\bar{x\\_i})}$, a path in the obtained net can visit at most two of these new cuts, but it reduces to the left hand side, with *ξ* such that $\\vc(\\xi)\\ge n$.\nIt might be possible to adapt our method for dealing with a relaxed definition of visited cut: for instance, we might consider the number of cuts *c* such that *ξ* visits a tree $t\\in\\ST(c)$ (instead of *t* ∈ *c*). But this notion is no longer local, and would introduce further technicalities: for that reason, we decided to focus on the length of paths instead, which is a more intuitive and standard notion, without any *ad hoc* reference to cuts.\nVariations of $\\jd(p)$ under reduction\n======================================\nFor establishing that $\\jd(q)$ is bounded as a function of $\\jd(p)$ and ln(*p*) we examine the reductions separately.\n[qigrowthm] Let *p*, *q* two proof nets. If $p\\totom q$, then $\\jd(q)\\leq\n 2\\jd(p)$.\nFor all $t\\in\\ST(q)$ the only case in which $\\jump^{-1}\\_q(t)\\not=\\jump^{-1}\\_p(t)$ is that of redirected jumps: there must be $\\w\\in\\counits(q)$ such that *t* is part of a residual $\\cut{t}{s}$ of an eliminated cut $\\cut{t\\_0}{s\\_0}\\in\\cuts(p)$, with $\\jump\\_p(\\w)=t\\_0$. In this case we have $\\jd\\_q(t)=\\jd\\_p(t)+\\jd\\_p(t\\_0)\\le 2\\jd(p)$.\n[qigrowthe] Let *p*, *q* two proof nets. If $p\\totoe q$, then $\\jd(q)\\leq\n (2\\jd(p))^{\\ln(p)+1}$.\nFix $t\\in\\ST(q)$. For any $\\w\\in\\counits(q)$, if $\\jump\\_q(\\w)=t$, then $\\w\\in\\I\\_p^n(t)$ for some $n\\in\\N$: this is precisely the purpose of the definition of $\\I^n\\_p$. We obtain $\\jd\\_q(t)\\le \\card\\left(\\bigcup\\_{n\\in\\N}\\I\\_p^n(t)\\right)\\leq (2\\jd(p))^{\\ln(p)+1}$ by Lemma [borneqi] (1).\n[qigrowtha] Let *p*, *q* two proof nets. If $p\\toto\\_a q$, then $\\jd(q)\\leq\n (\\ln(p)+1)\\jd(p)$.\nAs in the proof of Lemma [ax], we can write $p=(\\vec{c\\_1},\\dotsc,\\vec{c\\_k},\\vec c;\\vec s)$ and $q= (\\vec c;\\vec s)[t^1/\\bar x^1\\_0]\\cdots[t^k/\\bar x^k\\_0]$ where $\\vec c\\_i=(\\cut{x^i\\_0}{\\bar x^i\\_1},\\dotsc,\\cut{x^i\\_{n\\_i-1}}{\\bar x^i\\_{n\\_i}},\\cut{x^i\\_{n\\_i}}{t^i})$.\nBy the definition of cut elimination, for all $\\w\\in\\counits(q)$ we have: \n$$\\jump\\_q(\\w)= \\begin{cases}\n t^i [t^{i+1}/\\bar x\\_0^{i+1}]\\cdots[t^k/\\bar x\\_0^k]\n &\\text{if $\\jump\\_p(\\w)\\in\\{x^i\\_0,\\bar x^i\\_0,x^i\\_1,\\dots,\\bar x^i\\_{n\\_i},t^i\\}$}\n \\\\\n \\jump\\_q(\\w)[t^{1}/\\bar x\\_0^{1}]\\cdots[t^k/\\bar x\\_0^k]\n &\\text{otherwise}\n \\end{cases}.$$ \n It follows that:\n* $\\jd\\_q(t^i [t^{i+1}/\\bar x\\_0^{i+1}]\\cdots[t^k/\\bar x\\_0^k])=\\jd\\_p(t^i)+\\sum\\_{j=0}^{n\\_i}(\\jd\\_p(x^i\\_j)+\\jd\\_p(\\bar x^i\\_j))\n \\le (2n\\_i+3)\\jd(p)$;\n* if $t\\in\\ST(p)\\setminus\\bigcup\\_{i=1}^k\\{x^i\\_0,\\bar x^i\\_0,x^i\\_1,\\dots,\\bar x^i\\_{n\\_i},t^i\\}$, then $\\jd\\_q(t[t^{1}/\\bar x\\_0^{1}]\\cdots[t^k/\\bar x\\_0^k])=\\jd\\_p(t)$.\nTo conclude, it is sufficient to observe that 2*n**i* + 2 ≤ ln(*p*): indeed each $\\vec c\\_i$ induces a path alternating between *n**i* + 1 axioms an *n**i* + 1 cuts.\nBounding the size of antireducts: general and iterated case\n===========================================================\nThe previous results now allow us to treat the general case of a reduction $p\\toto q$.\n[sizebound] If $p\\toto q$ then $\\size(p)\\leq\n \\psi\\left(2(\\ln(p)+1)\\size(q),\\ln(p),\\jd(p)\\right)$.\nConsider *q*ʹ, *q*ʺ such that $p\\totoe q'\\totoa q''\\totom q$. We have:\n \n$$\\begin{aligned}\n \\size(p)\n &\\leq\\psi(\\size(q'),\\jd(p),\\ln(p))\n &&\\text{(by Lemma~\\ref{affs\\_size})}\\\\\n &\\leq\\psi((\\ln(q')+1)\\size(q''),\\jd(p),\\ln(p))\n &&\\text{(by Lemma~\\ref{ax})}\\\\\n &\\leq\\psi((\\ln(p)+1)\\size(q''),\\jd(p),\\ln(p))\n &&\\text{(by Lemma~\\ref{cc\\_ax\\_e})}\\\\\n &\\leq\\psi(2(\\ln(p)+1)\\size(q),\\jd(p),\\ln(p))\n &&\\text{(by Lemma~\\ref{mult})}\\quad.\n \\qedhere\n \\end{aligned}$$ \n[corollary:antireducts:onestep] If *q* is an MLL net and $n,m\\in\\N$, then \n$$\\{p\\tq p\\toto q,\\ \\jd(p)\\le m\\text{ and }\\ln(p)\\le n\\}$$ \n is finite.\nOf course, that result holds only up to the *α*-equivalence mentioned in Remark [rem:alpha]: then it is easy to check that the cardinality of $\\{p\\tq \\size(p)\\le k\\}$ is bounded by a function of $k\\in\\N$. Also recall from Remark [rem:jumps] that we are actually interested in bare nets rather than nets with jumps, so Corollary [corollary:antireducts:onestep] should be read as follows: given a bare net *q* and $n,m\\in\\N$ there are finitely many bare nets *p* such that $p\\toto q$ and that can be equipped with a jump function $\\jump\\_p$ satisfying $\\jd(p)\\le m$ and ln(*p*) ≤ *n*. More precisely, Theorem [sizebound] entails that the number of such bare nets *p* can be bounded by a function of *m*, *n* and $\\size(q)$.\nIt follows that, given an infinite linear combination ∑*i* ∈ *I**a**i*.*p**i*, assuming that we can equip each *p**i* with a jump function $\\jump\\_{p\\_i}$ so that $\\{\\ln(p\\_i)\\tq i\\in I\\}\\cup\\{\\jd(p\\_i)\\tq i\\in I\\}$ is finite, we can always consider an arbitrary family of reductions $p\\_i\\toto q\\_i$ for *i* ∈ *I* and form the sum ∑*i* ∈ *I**a**i*.*q**i*: this is always well defined. But if we want to iterate this process and perform a reduction from ∑*i* ∈ *I**a**i*.*q**i* to ∑*i* ∈ *I**a**i*.*r**i*, when $q\\_i\\toto r\\_i$ for *i* ∈ *I*, we need to ensure that a similar side condition holds for the *q**i*’s. Again, this is a consequence of our previous results, which we sum up in the following two theorems.\n[vcgrowth] Let $p\\toto q$. Then ln(*q*) ≤ *φ*(ln(*p*)).\nConsider *q*ʹ, *q*ʺ such that $p\\totom q'\\totoe q''\\totoa q$. We have:\n \n$$\\begin{aligned}\n \\ln(q)\n &\\leq \\ln(q'')\n &&\\text{(by Lemma~\\ref{cc\\_ax\\_e})}\\\\\n &\\leq \\ln(q')\n &&\\text{(by Lemma~\\ref{cc\\_ax\\_e})}\\\\\n &\\leq \\varphi(\\ln(p))\n &&\\text{(by Corollary~\\ref{vc})}\\quad.\n \\qedhere\n \\end{aligned}$$ \n[qigrowth] There exists a function $\\theta:\\N\\to\\N$ such that $\\jd(q)\\leq \\theta(\\ln(p),\\jd(p))$ whenever $p\\toto q$.\nConsider *q*ʹ, *q*ʺ such that $p\\totoa q'\\totoe q'' \\totom q$. We have\n \n$$\\begin{aligned}\n \\jd(q)\n &\\leq 2\\jd(q'')\n &&\\text{(by Lemma~\\ref{qi\\_growth\\_m})}\\\\\n &\\leq 2(2\\jd(q'))^{\\ln(q')+1}\n &&\\text{(by Lemma~\\ref{qi\\_growth\\_e})}\\\\\n &\\leq 2(2(\\ln(p)+1)\\jd(p))^{\\ln(q')+1}\n &&\\text{(by Lemma~\\ref{qi\\_growth\\_a})}\\\\\n &\\leq 2(2(\\ln(p)+1)\\jd(p))^{\\ln(p)+1}\n &&\\text{(by Lemma~\\ref{cc\\_ax\\_e})}\\quad.\n \\qedhere\n \\end{aligned}$$ \nBy the previous results, we can iterate Corollary [corollary:antireducts:onestep] and obtain:\n[corollary:antireducts] If *q* is an MLL net and $k,n,m\\in\\N$, then \n$$\\{p\\tq p\\toto^k q,\\ \\jd(p)\\le m\\text{ and }\\ln(p)\\le n\\}$$ \n is finite.\nTaylor expansion\n================\nWe now show how the previous results apply to Taylor expansion. For that purpose, we must extend our syntax to MELL proof nets. Our presentation departs from Ehrhard’s in our treatment of promotion boxes: instead of introducing boxes as tree constructors labelled by nets, with auxiliary ports as inputs, we consider box ports as 0-ary trees, that are related with each other in a *box context*, associating each box with its contents. This is in accordance with the usual presentation of promotion as a black box, and has two motivations:\n* in Ehrhard’s syntax, the promotion is not a net but an open tree, for which the trees associated with auxiliary ports must be mentioned explicitly: this would complicate the expression of Taylor expansion;\n* since we consider a single class of $\\wn$-links instead of having a separate dereliction, we must impose constraints on auxiliary ports, that are easier to express when these ports are directly represented in the syntax.\nThen we show that if *p* is a resource net in the support of the Taylor expansion of an MELL proof net *P*, then ln(*p*) and $\\jd(p)$ are bounded by functions of *P*.\nObserve that we need only to consider the support of Taylor expansion, so we do not formalize the expansion of MELL nets into infinite linear combinations of resource nets: rather, we introduce $\\taylor(P)$ as a set of approximants.\nMELL nets\n---------\nIn addition to the set of variables, we fix a denumerable set $\\fp$ of : we assume given an enumeration $\\fp=\\{a^b\\_i\\tq i,b\\in\\N\\}$. We call the ports *a*0*b* and the other ports. Instead of separate contractions and derelictions, we consider a unified $\\wn$-link of arbitrary arity; auxiliary ports of boxes must be premises of such links (or of auxiliary ports of outer boxes, that must satisfy this constraint inductively).\nThe weakenings (and coweakenings, in the resource nets yet to be introduced) are not essentially different from the multiplicative units in our untyped nets. Indeed, we will see that the geometrical and combinatorial behaviour of the $\\wn$-link (resp. the $\\oc$-link) is identical to that of the $\\parr$ (respectively, of the ⊗ ). This will be reflected in our use of labels: in addition to $\\units$ and $\\counits$, we will use labels from denumerable sets $\\coweaks$ and $\\weaks$ (now assuming $\\fv$, $\\fp$, $\\units$, $\\counits$, $\\coweaks$ and $\\weaks$ are pairwise disjoint), and write $\\pleaves=\\units\\cup\\coweaks$ and $\\nleaves=\\counits\\cup\\weaks$.\nWe introduce the corresponding term syntax. (*S*∘, *T*∘, *etc.*) and (*S*, *T*, *etc.*) are defined by mutual induction as follows:\n \n$$\\begin{aligned}\n T&\\recdef x \\mid \\unit\\cw \\mid \\counit\\w \\mid \\otimes(T\\_1,\\dotsc,T\\_n)\\mid \\parr(T\\_1,\\dotsc,T\\_n) \\mid a^b\\_0 \\mid \\weak{\\w'} \\mid \\wn(T^\\circ\\_1,\\dotsc,T^\\circ\\_n)\n \\\\\n T^\\circ & \\recdef T \\mid a^b\\_{i+1}\\end{aligned}$$ \nwhere *x* ranges over $\\fv$, $\\cw$ ranges over $\\units$, $\\w$ ranges over $\\counits$, $\\w'$ ranges over $\\weaks$, *b* and *i* range over $\\N$ and we require $n\\not=0$ in each of $\\otimes(T\\_1,\\dotsc,T\\_n)$, $\\parr(T\\_1,\\dotsc,T\\_n)$ and $\\wn(T^\\circ\\_1,\\dotsc,T^\\circ\\_n)$. The set $\\SPT(S)$ of the sub-pre-trees of *S* is defined in the natural way, as well as the set $\\ST(S)$ of sub-trees of *S*, from which we derive the definitions of $\\fv(S)$, $\\fp(S)$, $\\counits(S)$, *etc.* The set $\\atoms(S)$ of atoms of *S* is then $\\fv(S)\\cup\\fp(S)\\cup\\units(S)\\cup\\nleaves(S)$.\nA (resp. a ) is a raw tree (resp. raw pre-tree) in which each atom occurs at most once. A is an unordered pair of trees $C=\\cut{T}{S}$ with disjoints sets of atoms. Pre-trees and cuts only describe the surface level of MELL nets: we also have to introduce promotion boxes.\nWe now define and by mutual induction as follows. A box context Θ is the data of a finite set $\\boxes\\_{\\Theta}\\subset\\N$, and of a pre-net of the form $\\Theta(b)=(\\Theta\\_b;\\vec C\\_b;T\\_b,\\vec S^\\circ\\_b;\\jump\\_b)$, for each $b\\in\\boxes\\_{\\Theta}$. We then write $\\ar\\_\\Theta(b)$, or simply $\\ar(b)$ for the length of the family $\\vec S^\\circ\\_b$, which we call the of the box *b*. A pre-net is a tuple $P^\\circ=(\\Theta;\\vec C;\\vec S^\\circ;\\jump)$ where:\n* Θ is a box context;\n* the $\\jump$ is a function $\\nleaves(\\vec C,\\vec S^\\circ)\\to \\ST(\\vec C,\\vec S^\\circ)$;\n* each atom occurs at most once in $\\vec C,\\vec S^\\circ$;\n* $a^b\\_i\\in\\fp(\\vec C;\\vec S^\\circ)$ iff $b\\in\\boxes\\_\\Theta$ and $0\\le i\\le\\ar(b)$;\n* $\\fv(\\vec C,\\vec S^\\circ)$ is closed under the involution $x\\mapsto \\bar x$.\nThen a is a pre-net of the form $P=(\\Theta;\\vec C;\\vec S;\\jump)$, *i.e.* without auxiliary ports as conclusions. In the following, we may write, e.g., Θ*P* for Θ in this case. An example is illustrated in Figure [fig:exmell].\n[rem:mellnets] To be formal, in the definition of a box context Θ, we should also fix an enumeration of the family $\\vec S^\\circ\\_b$ in Θ(*b*). Indeed, when we write $a\\_0^b,a\\_1^b,\\dots,a\\_{\\ar(b)}^b$ for the ports of a box *b*, and $\\Theta(b)=(\\Theta\\_b;\\vec C\\_b;T\\_b,S^\\circ\\_{b,1},\\dots,S^\\circ\\_{b,\\ar(b)};\\jump\\_b)$ for the contents of the box, we implicitly assume a bijection which maps each pre-tree *S**b*, *i*∘ to the auxiliary port *a**i**b* of which it is a premise (which leaves *T* to be mapped to *a*0*b*). We prefer to keep this information implicit in the following, as the notations should allow to recover it, whenever necessary.\nOn the other hand, an analogue of Remark [rem:alpha] applies in this new setting, as pre-nets and nets should be considered up to some notion of isomorphism preserving the interface, which amounts to:\n* reindexing cuts, so that $\\vec C$ is considered as a set;\n* reordering premises of $\\wn$-links, which accounts for the associativity and commutativity of the underlying binary contraction;\n* renaming atoms and boxes and, simultaneously, changing the enumeration of the family $\\vec S^\\circ\\_b$ in each box, all this preserving the duality involution on variables, the partition $\\{a^b\\_i\\tq i\\in\\N\\}\\_{b\\in\\N}$ of $\\fp$, the jump functions, and the association of each *S**b*, *i* + 1∘ to *a**i* + 1*b*.\nWe still consider this as a form of *α*-equivalence as it only involves particular renamings of atoms or indices, preserving the rest of the structure. Again, we keep this quotient implicit whenever possible in the remaining.\nAlso, as already mentioned in Remark [rem:jumps], we rely on jumps to control the combinatorics of the elimination of evanescent cuts: we need nets to be equipped with jumps only to ensure that the resource nets in the Taylor expansion can also be equipped with jumps, that moreover enjoy uniform bounds. More precisely, we will show that if an MELL net *P* can be equipped with $\\jump\\_P$ that satisfies the acyclicity criterion, then each $p\\in\\taylor(P)$ can be equipped with $\\jump\\_p$, satisfying uniform bounds on ln(*p*) and $\\jd(p)$.\nThe existence of such a jump function $\\jump\\_P$ should be understood as side condition only: we keep it in the definition of nets by default because we rely on it everywhere in the following, but in the end we are actually interested in the compatibility of Taylor expansion with cut elimination for nets without jumps. And the reader may check that, without jumps, our pre-nets (up to *α*-equivalence) are essentially the same as, e.g., the in-PS’s (up to the names of internal ports) defined by de Carvalho for his proof of the injectivity of Taylor expansion.\n(ax) at (0,0)ax; (w) at ((*a**x*) + ( − 1.7, 0))$\\wn$; (b0)$\\oc$; (b01); (cut) at ((.5\\*(w)+.5\\*(b0)) + (0, − 0.7))cut; at (( − .7, .8) + (*b*0)) (ax0)ax; at (( − 1.7, 0) + (*a**x*0)) (bot0)⊥; (b02) at (bot0 |- b0); at (bot0 |- cut) (d0)$\\wn$; at ((.7, − 1.4) + (*a**x*)) (b)$\\oc$; at (( − 3.2, − .8) + (*b*)) (c)$\\wn$; (b1); (b2) at (b-|b01); at (d0 |- c) (d)$\\wn$; (b3) at (d0|-b);\n(x) at (ax0)*x* ; at (ax0)$\\ \\bar x$; (y) at (ax) *y* ; at (ax) $\\ \\bar y$; at (w.south)$\\w$; at (bot0)$\\w'$; at (b01)1; at (b02)2; at (b1)1; at (b2)2; at (b3)3; at (b)*b*; at (b0)*b*ʹ;\n(w) to[out=80,in=100] (y); (bot0) to[out=80,in=100] (x); (cut)to[out=west,in=south](b0); (cut)to[out=east,in=south](w); (ax0)to[out=east,in=north](b0); (ax)to[out=east,in=north](b);\n(in0) at (b0); (in) at (b);\n(bot0)–(bot0|-in0) (b02)–(d0)–(b3|-in) (b3)–(d) ; (ax0) to[out=west,in=north] (b01|-in0) (b01)–(b2|-in) (b2) to[out=south,in=north west] (c); (ax) to[out=west,in=north] (b1|-in) (b1) to[out=south,in=north east] (c); (b0.east) –++(.3,0) –++(0,1.7) –++(-3.8,0) –++(0,-1.7) – (b0.west); (b.east) –++(.3,0) –++(0,3.4) –++(-8.5,0) –+(0,-3.4) – (b.west); (c)–(c|- 0,-2.7); (b)–(b|- 0,-2.7); (d)–(d|- 0,-2.7);\n[fig:exmell]\nGiven a pre-net $P^\\circ=(\\Theta;\\vec C;\\vec S^\\circ;\\jump)$, we write $\\fv(P^\\circ)=\\fv(\\vec C;\\vec S^\\circ)$, $\\ST(P^\\circ)=\\ST(\\vec C;\\vec S^\\circ)$, *etc.* We define the of MELL pre-nets by $\\tlsize(P^\\circ)=\\card\\SPT(P^\\circ)$. We write $\\depth(P^\\circ)$ for the maximum level of nesting of boxes in *P*∘, *i.e.* the inductive depth in the above definition of pre-nets. The of MELL pre-nets includes that of their boxes: we set $\\size(P^\\circ)=\\tlsize(P^\\circ)+\\sum\\_{b\\in\\boxes\\_\\Theta}\\size(\\Theta(b))$ — this definition is of course by induction on $\\depth(P^\\circ)$.\nNotice that, by the above definition, for all $\\w\\in\\nleaves(\\vec C,\\vec S^\\circ)$, $\\jump\\_{P^\\circ}(\\w)$ must be at the same depth as $\\w$, and cannot be an auxiliary port.\nWe extend the switching functions of MLL to $\\wn$-links: for each $T=\\wn(T^\\circ\\_1,\\dots,T^\\circ\\_n)\\in\\ST(P^\\circ)$, *I*(*T*) ∈ {*T*1∘, …, *T**n*∘}, which induces a *T* ∼ *T**P*∘, *I**I*(*T*). We also consider *a*0*b* ∼ *b*, *i**P*∘*a**i**b* for $b\\in\\boxes\\_\\Theta$ and 1 ≤ *i* ≤ *a**r*(*b*): w.r.t. paths, a box *b* behaves like $\\ar(b)$ axiom links having the principal port of the box as a common vertice, and the content is not considered. Finally, also include the case of weakenings: $\\w\\sim^{P^\\circ}\\_{\\w} \\jump(\\w)$ for $\\w\\in\\nleaves(P^\\circ)$.\nWe write $\\paths(P^\\circ,I)$ (resp. $\\paths(P^\\circ)$) for the set of *I*-paths (resp. paths) in *P*∘. We say a pre-net *P*∘ is if there is no cycle in $\\paths(P^\\circ)$ and, inductively, each Θ(*b*) is acyclic. From now on, we consider acyclic pre-nets only.\nResource nets and Taylor expansion\n----------------------------------\nThe Taylor expansion of a net *P* will be a set of : these are the same as the multiplicative nets introduced before, with the addition of term constructors for $\\oc$ and $\\wn$. Raw trees are given as follows: \n$$t\\recdef x \\mid \\unit\\cw \\mid\\counit\\w\\mid \\otimes(t\\_1,\\dotsc,t\\_n)\\mid \\parr(t\\_1,\\dotsc,t\\_n)\\mid \\coweak{\\cw'}\\mid \\weak{\\w'} \\mid \\oc(t\\_1,\\dotsc,t\\_n) \\mid \\wn(t\\_1,\\dotsc,t\\_n).$$ \n where *x* ranges in $\\fv$, $\\cw$ ranges over $\\units$, $\\w$ ranges over $\\counits$, $\\cw'$ ranges over $\\coweaks$, $\\w'$ ranges over $\\weaks$, and we require $n\\not=0$ in each case. In resource nets, we extend switchings to $\\wn$-links and jumps from weakenings as in MELL nets, associated with $\\wn$-edges and jump edges. Moreover, for each $t=\\oc(t\\_1,\\dots,t\\_n)$, we have *t* ∼ *t*, *t**i**t**i* for 1 ≤ *i* ≤ *n*. Observe that, except for the notation of the root constructor, the trees $\\coweak{\\cw}$, $\\weak{\\w}$, $\\oc(t\\_1,\\dotsc,t\\_n)$ and $\\wn(t\\_1,\\dotsc,t\\_n)$, are exactly the same as $\\unit\\cw$, $\\counit\\w$, $\\otimes(t\\_1,\\dotsc,t\\_n)$ and $\\parr(t\\_1,\\dotsc,t\\_n)$ respectively: in particular they induce the same geometry for paths.\nDuring Taylor expansion, we need to replace a box in a pre-net with an arbitrary number of approximants of this box. Let us call the data $r=(\\vec s\\_0,\\dotsc,\\vec s\\_n)$ of *n* + 1 families of pairwise distinct resource trees $\\vec s\\_0,\\dotsc,\\vec s\\_n$, such that $\\atoms(s)\\cap\\atoms(s')=\\emptyset$ whenever $s\\in\\vec s\\_i$ and $s'\\in\\vec s\\_{i'}$, and $i\\not=i'$ or $s\\not=s'$. A family $\\vec r=(r\\_b)\\_{b\\in \\boxes}$ of box replacements such that $\\atoms(r\\_b)\\cap\\atoms(r\\_{b'})=\\emptyset$ for $b\\not=b'\\in\\boxes$ is if $\\atoms(T^\\circ)\\cap\\atoms(\\vec r)=\\emptyset$ and, for each $a^b\\_i\\in\\fp(T^\\circ)$, $b\\in\\boxes$ and *r**b* is of arity at least *i*.\n[replacements] Let $\\vec r$ be a *B*-indexed family of box replacements, and write $r\\_b=(\\vec s^b\\_0,\\dotsc,\\vec s^b\\_{n\\_b})$ for each *b* ∈ *B*. Assuming that $\\vec r$ is applicable to the tree *S* (resp. the pre-tree *S*∘), the (resp. of *S*∘) is the tree $\\bsubs{S}{\\vec r}$ (resp. the family of pre-trees $\\rsubs{S^\\circ}{\\vec r}$) defined by mutual induction on pre-trees and trees as follows:\n \n$$\\begin{gathered}\n \\begin{aligned}\n \\bsubs{x}{\\vec r}&=x&\n \\bsubs{\\cw}{\\vec r}&=\\cw&\n \\bsubs{\\w}{\\vec r}&=\\w&\n \\bsubs{a^b\\_0}{\\vec r}&=\\begin{cases}\n \\cw\\_b&\\text{if $\\vec s^b\\_0$ is empty}\\\\\n \\oc(\\vec s^b\\_0)&\\text{otherwise}\\\\\n \\end{cases}\n \\end{aligned}\n \\\\\n \\begin{aligned}\n \\bsubs{\\otimes(T\\_1,\\dotsc,T\\_n)}{\\vec r}&=\n \\otimes(\\bsubs{T\\_1}{\\vec r},\\dotsc,\\bsubs{T\\_n}{\\vec r})\\\\\n \\bsubs{\\parr(T\\_1,\\dotsc,T\\_n)}{\\vec r}&=\n \\parr(\\bsubs{T\\_1}{\\vec r},\\dotsc,\\bsubs{T\\_n}{\\vec r})\\\\\n \\bsubs{\\wn(T^\\circ\\_1,\\dotsc,T^\\circ\\_n)}{\\vec r}&=\n \\begin{cases}\n \\w\\_{\\wn(T^\\circ\\_1,\\dotsc,T^\\circ\\_n)}&\n \\text{if $\\rsubs{T^\\circ\\_1}{\\vec r},\\dotsc,\\rsubs{T^\\circ\\_n}{\\vec r}$ is empty}\\\\\n \\wn(\\rsubs{T^\\circ\\_1}{\\vec r},\\dotsc,\\rsubs{T^\\circ\\_n}{\\vec r})&\n \\text{otherwise}\n \\end{cases}\n \\end{aligned}\n \\\\\n \\rsubs{T}{\\vec r}=\\bsubs{T}{\\vec r}\\qquad\n \\rsubs{a^b\\_{i+1}}{\\vec r}=\\vec s^b\\_{i+1}\n \\end{gathered}$$ \nwhere each $\\cw\\_b\\in\\coweaks$ and each $\\w\\_{\\wn(T^\\circ\\_1,\\dotsc,T^\\circ\\_n)}\\in\\weaks$ is chosen fresh (not in $\\atoms(S)$ nor $\\atoms(S^\\circ)$ nor $\\atoms(\\vec r)$) and unique.[10](#fn10)\nWe are now ready to introduce the expansion of MELL nets depicted in Figure [fig:taylor].[11](#fn11) During the construction, we need to track the conclusions of copies of boxes, in order to collect copies of auxiliary ports in the external $\\wn$-links: this is the role of the intermediate notion of pre-Taylor expansion.\nFirst, recall that we write $T\\_b,S^\\circ\\_{b,1},\\dots,S^\\circ\\_{b,\\ar(b)}$ for the trees of Θ(*b*) that are respectively mapped to $a\\_0^b,a\\_1^b,\\dots,a\\_{\\ar(b)}^b$. Also, in this case, let us write $\\vec S^\\circ\\_b=(S^\\circ\\_{b,1},\\dotsc,S^\\circ\\_{b,\\ar(b)})$.\n[defTaylor] Given a closed pre-net $P^\\circ=(\\Theta;\\vec C;\\vec S^\\circ;\\jump)$, a of *P*∘ is any pair (*p*, *f*) of a resource net $p=(\\vec c;\\vec t;\\jump\\_p)$, together with a function $f:\\vec t\\to\\vec S^\\circ$ such that *f*− 1(*T*) is a singleton whenever $T\\in\\vec S^\\circ$ is a tree, obtained as follows:\n* for each $b\\in\\boxes\\_{\\Theta}$, fix a number *k**b* ≥ 0 of copies;\n* for 1 ≤ *j* ≤ *k**b*, fix inductively a pre-Taylor expansion (*p**j**b*, *f**j**b*) of Θ(*b*), renaming the atoms so that the sets $\\atoms(p^b\\_j)$ are pairwise disjoint, and also disjoint from $\\atoms(\\vec C)\\cup\\atoms(\\vec S^\\circ)$;\n* write $p^b\\_j=(\\vec c^b\\_j;t^b\\_j,\\vec s^b\\_j;\\jump^b\\_j)$ so that *f**j**b*(*t**j**b*) = *T**b*;\n* write $\\vec r=(r\\_b)\\_{b\\in \\boxes\\_\\Theta}$ for the family of box replacements $r\\_b=(\\vec u^b\\_0,\\dotsc,\\vec u^b\\_{\\ar(b)})$, where $\\vec u^b\\_0=(t^b\\_1,\\dotsc,t^b\\_{k\\_b})$ and each $\\vec u^b\\_i$ is an enumeration of ⋃*j* = 1*k**b*(*f**j**b*)− 1(*S**b*, *i*∘) for $1\\le i\\le \\ar(b)$;\n* set $\\vec t=\\rsubs{\\vec S^\\circ}{\\vec r}$ and $\\vec c=\\bsubs{\\vec C}{\\vec r},\\vec c'$ where $\\vec c'$ is the concatenation of the families $\\vec c^b\\_j$ for $b\\in \\boxes\\_\\Theta$ and 1 ≤ *j* ≤ *k**b*\n* for $t\\in\\vec t$, set *f*(*t*) = *a**i**b* if $t\\in\\vec u^b\\_i$ with $1\\le i\\le \\ar(b)$, otherwise let *f*(*t*) be the tree $T\\in\\vec S^\\circ$ such that $t=\\bsubs T{\\vec r}$;\n* for each $\\w\\in\\nleaves(p)$, $\\jump\\_p(\\w)$ is defined as follows:\n\t+ if $\\w\\in\\nleaves(p\\_j^b)$ then we set $\\jump\\_p(\\w)=\\jump\\_{p\\_j^b}(\\w)$.\n\t+ if $\\w=\\w\\_{\\wn(T^\\circ\\_1,\\dotsc,T^\\circ\\_n)}$ then each $T\\_i^\\circ\\{\\vec r\\}$ is empty; then we select any $i\\in\\{1,\\dotsc, n\\}$ and set $\\jump\\_p(\\w)=\\bsubs{a\\_0^b}{\\vec r}$ where $b\\in\\boxes\\_\\Theta$ is the box such that *T**i*∘ = *a**j**b* for some $1\\le j\\le \\ar(b)$;\n\t+ otherwise $\\w\\in\\nleaves(\\vec C;\\vec S^\\circ)$, and then we set $\\jump\\_p(\\w)=\\bsubs{\\jump(\\w)}{\\vec r}$ (note that $\\jump(\\w)$ is a tree so this is a valid application of Definition [replacements]).\nThe of a net *P* is then $\\taylor(P)=\\{p\\tq (p,f)\\text{ is a pre-Taylor expansion of }P\\}$.\nGiven the net $P=(\\Theta;;\\wn(a^b\\_3),\\wn(a^b\\_2,a^b\\_1),a^b\\_0;\\jump)$ of Figure [fig:exmell], we construct an element *p* of $\\taylor(P)$ as follows. First, we take two copies of box *b*, fixing *k**b* = 2. Recall that $\\Theta(b)=(\\Theta';\\cut{a^{b'}\\_0}{\\w};\\bar y,y,a^{b'}\\_1,\\wn(a^{b'}\\_2);\\jump')$ where $\\boxes\\_{\\Theta'}=\\{b'\\}$. Hence, to construct (*p**j**b*, *f**j**b*) we must first fix a number *k**b*ʹ, *j* of copies of the box *b*ʹ: we set *k**b*ʹ, 1 = 0 and *k**b*ʹ, 2 = 1, and it remains to select a single pre-Taylor expansion (*p*ʹ, *f*ʹ) of Θʹ(*b*ʹ) for the only copy of *b*ʹ in (*p*2*b*, *f*2*b*). Since $\\Theta'(b')=(\\Theta'';;\\bar x,x,\\w';\\jump'')$ contains no box, we must have $p'=(;\\bar x,x,\\w';\\jump'')$ with $f'(\\bar x)=\\bar x$, *f*ʹ(*x*) = *x* and $f'(\\w')=\\w'$.\nSince *k**b*ʹ, 1 = 0, we construct $p^b\\_1=\\big(\\rsubs{\\cut{a^{b'}\\_0}{\\w}}{r\\_{b',1}};\\rsubs{(\\bar y,y,a^{b'}\\_1,\\wn(a^{b'}\\_2))}{r\\_{b',1}};\\jump^b\\_1\\big)$ where *r**b*ʹ, 1 is the empty replacement: we obtain $p^b\\_1=(\\cut{\\cw}{\\w};\\bar y,y,\\w'')$ where $\\cw\\in\\coweaks$ and $\\w''\\in\\weaks$ are fresh, and we set $f^b\\_1(\\bar y)=\\bar y$, *f*1*b*(*y*) = *y*, $f^b\\_1(\\w'')=\\wn(a^{b'}\\_2)$, and also $\\jump^b\\_1(\\w)=\\bsubs{\\jump'(\\w)}{r\\_{b',1}}=y$ and $\\jump^b\\_1(\\w'')=\\bsubs{a^{b'}\\_0}{r\\_{b',1}}=\\cw$.\nHaving defined (*p*ʹ, *f*ʹ) as above, we must set $r\\_{b',2}=((\\bar x),(x),(\\w'))$ and we define $p^b\\_2=\\big(\\rsubs{\\cut{a^{b'}\\_0}{\\w}}{r\\_{b',2}};\\rsubs{(\\bar y,y,a^{b'}\\_1,\\wn(a^{b'}\\_2))}{r\\_{b',2}};\\jump^b\\_2\\big)$: we obtain $p^b\\_2=(\\cut{\\oc(\\bar x)}{\\w};\\bar y,y,x,\\wn(\\w'))$, with $f^b\\_2(\\bar y)=\\bar y$, *f*2*b*(*y*) = *y*, *f*2*b*(*x*) = *a*1*b*ʹ and $f^b\\_2(\\wn(\\w'))=\\wn(a^{b'}\\_2)$, and we set $\\jump^b\\_2(\\w)=\\bsubs{\\jump'(\\w)}{r\\_{b',2}}=y$ and $\\jump^b\\_2(\\w')=\\jump''(\\w')=x$.\nWe rename the atoms in both pre-Taylor expansions systematically as follows: $p^b\\_1=(\\cut{\\cw\\_1}{\\w\\_1};\\bar y\\_1,y\\_1,\\w''\\_1)$ and $p^b\\_2=(\\cut{\\oc(\\bar x\\_2)}{\\w\\_2};\\bar y\\_2,y\\_2,x\\_2,\\wn(\\w'\\_2))$, also redefining *f*1*b*, $\\jump^b\\_1$, *f*2*b* and $\\jump^b\\_2$ accordingly.\nFinally, we set $\\vec c=\\cut{\\cw\\_1}{\\w\\_1},\\cut{\\oc(\\bar x\\_2)}{\\w\\_2}$ and $\\vec t=\\rsubs{(\\wn(a^b\\_3),\\wn(a^b\\_2,a^b\\_1),a^b\\_0)}{r\\_b}$ where $r\\_b=((\\bar y\\_1,\\bar y\\_2),(y\\_1,y\\_2),(x\\_2),(\\w''\\_1,\\wn(\\w'\\_2)))$. We obtain: \n$$p=(\\cut{\\cw\\_1}{\\w\\_1},\\cut{\\oc(\\bar x\\_2)}{\\w\\_2};\\wn(\\w''\\_1,\\wn(\\w'\\_2)),\\wn(x\\_2,y\\_1,y\\_2),\\oc(\\bar y\\_1,\\bar y\\_2);\\jump\\_p)$$ \n with $\\jump\\_p(\\w\\_1)=y\\_1$, $\\jump\\_p(\\w\\_2)=y\\_2$, $\\jump\\_p(\\w''\\_1)=\\cw\\_1$ and $\\jump\\_p(\\w'\\_2)=x\\_2$, which is depicted in Figure [fig:extaylor].\n(coc)$\\oc$; at (( − 3, − .8) + (*c**o**c*)) (con)$\\wn$;\n(ax1) at ((*c**o**c*) + ( − 1, 1.5))ax; (w1) at ((*a**x*1) + ( − 2.2, 0))$\\wn$; (oc1)$\\oc$; (cut1) at ((.5\\*(w1)+.5\\*(oc1)) + (0, − 0.7))cut; at (( − 4, 0) + (*c**u**t*1)) (w11)$\\wn$; at (w11)$\\w''\\_1\\ $; (y1) at (ax1) *y*1 ; at (ax1) $\\ \\bar y\\_1$; at (w1.south)$\\w\\_1$; (cw1) at (oc1)$\\cw\\_1$;\n(w1) to[out=80,in=100] (y1); (w11) to[out=north east,in=north west] (oc1); (cut1)to[out=west,in=south](oc1); (cut1)to[out=east,in=south](w1); (ax1)to[out=east,in=105](coc);\n(ax2) at ((*a**x*1) + (0, 2)) ax; (w2) at (ax2 -| w1)$\\wn$; (oc2) at (ax2 -| oc1)$\\oc$; at (( − .7, .7) + (*o**c*2)) (ax20)ax; (cut2) at ((.5\\*(w2)+.5\\*(oc2)) + (0, − 0.7))cut; at (( − 1.7, 0) + (*a**x*20)) (bot2)⊥; at (bot2 |- cut2) (d2)$\\wn$; at (.5 \\* ((d2|-con)+(w11|-con))) (con’)$\\wn$;\n(y2) at (ax2) *y*2 ; at (ax2) $\\ \\bar y\\_2$; at (w2.south)$\\w\\_2$; (x2) at (ax20)*x*2 ; at (ax20)$\\ \\bar x\\_2$; at (bot2)$\\w'\\_2\\ $;\n(b21); (b2) at (ax1-|b21); (b1);\n(w2) to[out=80,in=100] (y2); (bot2) to[out=80,in=100] (x2); (cut2)to[out=west,in=south](oc2); (cut2)to[out=east,in=south](w2); (ax20)to[out=east,in=north](oc2); (ax20) to[out=west,in=north] (b21) to[out=south,in=north] (b2) to[out=south,in=north west] (con); (ax2) to[out=west,in=north] (b1) to[out=south,in=55] (con); (ax2)to[out=east,in=75](coc); (bot2)to[out=south,in=north](d2);\n(ax1) to[out=west,in=75] (con);\n(d2)to[out=south,in=75](con’); (w11)to[out=south,in=105](con’);\n(con) –(con |- 0,-1.3); (con’)–(con’|- 0,-1.3); (coc) –(coc |- 0,-1.3);\n[fig:extaylor]\nPaths in Taylor expansion\n-------------------------\nIn the following, we fix a pre-Taylor expansion (*p*, *f*) of $P^\\circ=(\\Theta;\\vec C;\\vec S^\\circ;\\jump)$ and we describe the structure of paths in *p*. We show that the critical case depicted in Figure [fig:cheminstaylor] is maximal, so that a path of *p* passes through at most two copies of each box of *P*∘.\n[scale=.6,baseline=(cdots.base)] (0.5,0)–(0.5,1)–(4.5,1)–(4.5,0)–cycle; (2.2,-0.3)node⋯; (2.5,0.5)node*p*1*b*;\n(5,0.5)node⋯; (5.5,0)–(5.5,1)–(9.5,1)–(9.5,0)–cycle; (7.5,0.5)node*p**k**b**b*; (6.8,-0.3)node⋯;\n(c1) at (3.5,-1.75)?; (c2) at (5.5,-1.75)?; (cc) at (6.5,-1.75)!; ; ; ; (cdots) at (4.5,-1.75) ⋯;\n(c1)–(3.5,-2.5); (c2)–(5.5,-2.5); (cc)–(6.5,-2.5); (1,0) to [out=270,in=125] (c1); (3,0) to [out=270,in=125] (c2); (4,0) to [out=270,in=125] (cc); (6,0) to [out=270,in=55] (c1); (8,0) to [out=270,in=55] (c2); (9,0) to [out=270,in=55] (cc); (0.8,0) to [out=270,in=125] ((*c*1) + ( − 0.5, 0)) to ((*c*1) + ( − 0.5, − 1)); (0.8,0) to[out=90,in=90] (4.2,0); (4.2,0) to[out=-90,in=125] (6.5,-1.75) to[out=55,in=-90] (9.2,0); (9.2,0)to[out=90,in=90] (7.8,0); (7.8,0) to[out=-90,in=55] ((*c*2) + ( − 0.5, 0)) to ((*c*2) + ( − 0.5, − 1));\n[fig:cheminstaylor]\nObserve that \n$$\\ST(p)=\n \\{\\bsubs{T}{\\vec r}\\tq T\\in\\ST(\\vec C,\\vec S^\\circ)\\}\n \\cup\\bigcup\\_{b\\in\\boxes\\_\\Theta}\\bigcup\\_{j=1}^{k\\_b} \\ST(p^b\\_j)$$ \n (using the notations of Definition [defTaylor]). It follows that, for each $t\\in\\ST(p)$:\n* either *t* is in a copy of a box, *i.e.* (up to *α*-equivalence) $t\\in\\ST(p^b\\_j)$ for some $b\\in\\boxes\\_\\Theta$ and 1 ≤ *j* ≤ *k**b*, and then we say *t* is and write $\\boxname(t)=b$ and $\\boxcopy(t)=(b,j)$;\n* or there exists a unique $T\\in\\ST(P^\\circ)$ such that $t=\\bsubs{T}{\\vec r}$, and then we say *t* is, and write $\\untaylor t=T$.\nWe further distinguish the of *p*, *i.e.* the outer trees $\\oc(t\\_1^b,\\dotsc,t\\_{k\\_b}^b)$ for $b\\in\\boxes\\_\\Theta$, which we denote by $\\oc\\_b$, so that $\\untaylor{\\oc\\_b}=a^b\\_0$.\nWe say an edge *t* ∼ *e**p*, *I**s* of *p* is an (resp. an ) if *t* and *s* are both inner (resp. outer) trees. We say a path $\\xi\\in\\paths(p)$ is an (resp. an ) if it crosses inner edges (resp. outer edges) only.\nIf *t* ∼ *e**p*, *I**s* is an inner edge then $\\boxcopy(t)=\\boxcopy(s)$ and we also have *t* ∼ *e**p**j**b*, *I**j**b**s* where $(b,j)=\\boxcopy(t)$ and *I**j**b* is the restriction of *I* to $\\nleaves(p^b\\_j)$. In this case, we also set $\\boxname(e)=b$ and $\\boxcopy(e)=(b,j)$. If *ξ* is an inner path, we set $\\boxname(\\xi)$ (resp. $\\boxcopy(\\xi)$) for the common value of $\\boxname$ (resp. $\\boxcopy$) on the edges crossed by *ξ*, and we obtain:\n[lemma:inner] If *ξ* is an inner path then $\\xi\\in\\paths(p^b\\_j,I^b\\_j)$ where $(b,j)=\\boxcopy(\\xi)$.\nThe classification of the outer edges of *p* is more delicate. First, we associate a switching $\\untaylor I$ of *P*∘ with each switching *I* of *p* as follows:\n* if $I(\\parr(\\bsubs{T\\_1}{\\vec r},\\dotsc,\\bsubs{T\\_n}{\\vec r}))=\\bsubs{T\\_i}{\\vec r}$, we set $\\untaylor I(\\parr(T\\_1,\\dotsc,T\\_n))=T\\_i$;\n* if $I(\\wn(\\rsubs{T^\\circ\\_1}{\\vec r},\\dotsc,\\rsubs{T^\\circ\\_n}{\\vec r}))\n \\in\\rsubs{T^\\circ\\_i}{\\vec r}$, we set $\\untaylor I(\\wn(T^\\circ\\_1,\\dotsc,T^\\circ\\_n))=T^\\circ\\_i$;\n* if $\\bsubs{\\wn(T^\\circ\\_1,\\dotsc,T^\\circ\\_n)}{\\vec r}=\\w$, *T**i*∘ = *a**j**b* and $\\jump\\_p(\\w)=\\oc\\_b$, we set $\\untaylor I(\\wn(T^\\circ\\_1,\\dotsc,T^\\circ\\_n))=T^\\circ\\_i$.[12](#fn12)\nIf *t* ∼ *e**p*, *I**s* is an outer edge then, in each of the following cases, we can define an $\\untaylor I$-edge $\\untaylor e$ of *P*∘ such that $e=\\bsubs{\\untaylor e}{\\vec r}$ and $\\untaylor t\\sim^{P^\\circ,\\untaylor I}\\_{\\untaylor e} \\untaylor s$:\n* *e* is an axiom edge, and we set $\\untaylor e=e$;\n* *e* is a ⊗ -edge, e.g. $t=\\otimes(t\\_1,\\dotsc,t\\_n)$ and *s* = *t**i*, and we set $\\untaylor e=(\\untaylor t,\\untaylor{t\\_i})$;\n* *e* is a $\\parr$-edge or a $\\wn$-edge, e.g. $t=\\parr(t\\_1,\\dotsc,t\\_n)$, *s* = *t**i* and *I*(*t*) = *s*, and we set $\\untaylor e=\\untaylor t$;\n* $e=\\cut ts$ is a cut edge, and we set $\\untaylor e=\\cut{\\untaylor t}{\\untaylor s}$;\n* $e=\\w\\in\\nleaves(P^\\circ)\\subset\\nleaves(p)$ and we set $\\untaylor e=e$ (observe that in this case we have $\\jump\\_{P^\\circ}(\\w)=\\untaylor{\\jump\\_p(\\w)}$).\nIf any of the above cases holds, we say the outer edge *e* is.\nIf *e* is an outer edge that is not superficial then *e* must be a : $e=\\w\\in\\weaks(p)\\setminus\\big(\\weaks(P^\\circ)\n \\cup\\bigcup\\_{b\\in\\boxes\\_\\Theta}\\bigcup\\_{j=1}^{k\\_b} \\weaks(p^b\\_j)\\big)$. If, e.g., $t=\\w$, then we can write $\\untaylor t=\\wn(T^\\circ\\_1,\\dotsc,T^\\circ\\_n)$ and $s=\\jump\\_p(t)=\\oc\\_b$ where *b* is such that $\\untaylor I(\\untaylor{t})=a^b\\_{i}$ with $1\\le i\\le \\ar(b)$. In this case, we obtain a path $\\boxedge e=\\untaylor t\\sim^{P^\\circ,\\untaylor I}\\_{\\untaylor t} a^b\\_i\n\\sim^{P^\\circ,\\untaylor I}\\_{b,i} a^b\\_0=\\untaylor s$.\n[lem:outer] If *ξ* is an outer *I*-path in *p*, then there exists an $\\untaylor I$-path $\\untaylor\\xi$ in *P*∘ with $\\ln(\\untaylor\\xi)\\ge\\ln(\\xi)$.\nIt is sufficient to replace each outer edge *e* crossed by *ξ* with:\n* either $\\untaylor e$ if *e* is superficial,\n* or the path $\\boxedge e$ or $\\reverse{\\boxedge e}$ if *e* is a created jump.\nObserve indeed that if *t* ∼ *e**p*, *I**s* and *t*ʹ ∼ *e*ʹ*p*, *I**s*ʹ are outer paths of length 1 with $e\\not=e'$ then the paths $\\untaylor{(t\\sim^{p,I}\\_e s)}:\\untaylor t\\leadsto\\_{P^\\circ,\\untaylor I} \\untaylor s$ and $\\untaylor{(t'\\sim^{p,I}\\_{e'} s')}:\\untaylor {t'}\\leadsto\\_{P^\\circ,\\untaylor I} \\untaylor {s'}$ thus defined are disjoint, and of length at least 1.\nSome edges are neither inner nor outer: a is an edge *t* ∼ *e**p*, *I**s* such that *t* is outer and *s* is inner, in which case we set $\\boxcopy(e)=\\boxcopy(s)$. There are two kinds of boundary edges:\n* the of the box copy (*b*, *j*) is the $\\oc$-edge $(\\oc\\_b,t^b\\_j)$;\n* an *e* of the box copy (*b*, *j*) is any $\\wn$-edge *t* ∼ *t**p*, *I**s* where $I(t)=s\\in\\vec s^b\\_j$ is such that *f**j**b*(*s*) = *S**b*, *i*∘ with $1\\le i\\le \\ar(b)$, in which case we must have $\\untaylor I(\\untaylor t)=a^b\\_i$, and then we write $\\boxdoor e=i$ for the index of the corresponding auxiliary port.\nWe call any path of the form *χ* = (*e*)*ξ*(*e*ʹ) where *e* and *e*ʹ are boundaries and *ξ* is an inner path: in this case, we write $\\boxname(\\chi)=\\boxname(\\xi)$ and $\\boxcopy(\\chi)=\\boxcopy(\\xi)$. Obviously, any path *ξ* with outer endpoints is obtained as an alternation of outer paths and box paths: we can write uniquely *ξ* = *ξ*0*χ*1*ξ*1⋯*χ**n**ξ**n* where each *ξ**i* is an outer path, and each *χ**i* is a box path.\nLet $\\chi=(e)\\xi(e'):t\\leadsto\\_p s$ be a box path with $\\boxcopy(\\chi)=(b,j)$. Since $e\\not=e'$, then at most one of *e* and *e*ʹ is the principal boundary, and if both *e* and *e*ʹ are auxiliary boundaries, then we must have $\\boxdoor e\\not=\\boxdoor{e'}$: indeed *e* is the only $\\wn$-edge whose premises include $\\rsubs{a^b\\_{\\boxdoor e}}{\\vec r}$. We can thus define $\\untaylor\\chi:\\untaylor t\\leadsto\\_{P^\\circ} \\untaylor s$ as follows:\n* if *e* and *e*ʹ are auxiliary boundaries then $\\untaylor\\chi=\\untaylor t\\sim\\_{\\untaylor t} a^b\\_{\\boxdoor e} \\sim\\_{b,\\boxdoor e}\n a^b\\_0\\sim\\_{b,\\boxdoor{e'}} a^b\\_{\\boxdoor{e'}} \\sim\\_{\\untaylor s} \\untaylor s$\n* otherwise, e.g. *e*ʹ is principal and we set $\\untaylor\\chi=\\untaylor t\\sim\\_{\\untaylor t} a^b\\_{\\boxdoor e} \\sim\\_{b,\\boxdoor e} a^b\\_0 =\\untaylor s$.\n[lem:cheminsTaylor] Assume $\\xi=\\xi\\_0\\chi\\_1\\xi\\_1\\cdots\\chi\\_n\\xi\\_n: t\\leadsto\\_{p,I} s$ where each *ξ**i* is an outer path, and each *χ**i* is a box path. Then, setting $\\untaylor\\xi=\\untaylor{\\xi\\_0}\\untaylor{\\chi\\_1}\\untaylor{\\xi\\_1}\\cdots\\untaylor{\\chi\\_n}\\untaylor{\\xi\\_n}$ we obtain $\\untaylor\\xi:\\untaylor t\\leadsto\\_{P^\\circ,\\untaylor I} \\untaylor s$. Moreover, if $\\boxname(\\chi\\_i)=\\boxname(\\chi\\_j)=b$ and *i* < *j*, then *j* = *i* + 1, $\\xi\\_i=\\epsilon\\_{\\oc\\_b}$, and $\\boxcopy(\\chi\\_i)\\not=\\boxcopy(\\chi\\_j)$.\nWe have already observed in the proof of Lemma [lem:outer] that if *ξ* and *ξ*ʹ are disjoint outer paths then $\\untaylor{\\xi}$ and $\\untaylor{\\xi'}$ are also disjoint. Similarly, if *ξ* is outer and *χ* is a box path, it follows directly from the definitions that $\\untaylor{\\xi}$ and $\\untaylor{\\chi}$ are disjoint. And if *χ* and *χ*ʹ are box paths with disjoint boundaries, again $\\untaylor{\\chi}$ and $\\untaylor{\\chi'}$ are disjoint paths by construction. It follows that, if $\\xi=\\xi\\_0\\chi\\_1\\xi\\_1\\cdots\\chi\\_n\\xi\\_n: t\\leadsto\\_{p,I} s$, each *ξ**i* is an outer path, and each *χ**i* is a box path, then the concatenation $\\untaylor\\xi=\\untaylor{\\xi\\_0}\\untaylor{\\chi\\_1}\\untaylor{\\xi\\_1}\\cdots\\untaylor{\\chi\\_n}\\untaylor{\\xi\\_n}$ is well defined.\nWrite $\\chi\\_i=(e\\_i)\\xi'\\_i(e'\\_i):t\\_i\\leadsto s\\_i$ for 1 ≤ *i* ≤ *n*. Assume $\\boxname(\\chi\\_i)=\\boxname(\\chi\\_j)=b$, and moreover $\\boxname(\\chi\\_k)\\not=b$ for *i* < *k* < *j*. We obtain a path $\\xi'=\\untaylor{(\\xi\\_i\\chi\\_{i+1}\\cdots\\xi\\_{j-1})}:\\untaylor{s\\_i}\\leadsto\\_{P^\\circ}\\untaylor{t\\_j}$: by construction, *ξ*ʹ does not cross any box edge (*b*, *l*) for $1\\le l\\le\\ar(b)$. If (*e*ʹ*i*) and (*e**j*) were both auxiliary, we could form a cycle $\\xi'(\\untaylor{t\\_j}\\sim\\_{\\untaylor{t\\_j}}a^b\\_{\\boxdoor{e\\_j}}\\sim\\_{b,\\boxdoor{e\\_j}}\n a^b\\_0\\sim\\_{b,\\boxdoor{e'\\_i}} a^b\\_{\\boxdoor{e'\\_i}} \\sim\\_{\\untaylor{s\\_i}}\\untaylor{s\\_i})$, since *ξ*ʹ would cross neither $\\untaylor{t\\_j}$ nor $\\untaylor{s\\_i}$. If, e.g., (*e*ʹ*i*) was principal and (*e**j*) was auxiliary, we could form a cycle $\\xi'(\\untaylor{t\\_j}\\sim\\_{\\untaylor{t\\_j}}a^b\\_{\\boxdoor{e\\_j}}\\sim\\_{b,\\boxdoor{e\\_j}} a^b\\_0)$, as *ξ*ʹ would not cross $\\untaylor{t\\_j}$. So both must be principal and we have $\\untaylor{s\\_i}=\\untaylor{t\\_j}=a^b\\_0$: since *P*∘ has no non empty cycle, we must have *ξ*ʹ = *ε**a*0*b* hence $\\xi\\_i\\chi\\_{i+1}\\cdots\\xi\\_{j-1}=\\epsilon\\_{\\oc\\_b}$ and then *j* = *i* + 1 and $\\xi\\_i=\\epsilon\\_{\\oc\\_b}$. Since $e'\\_i\\not=e\\_j$, we moreover obtain $\\boxcopy(\\chi\\_i)\\not=\\boxcopy(\\chi\\_j)$.\nIt remains only to prove that, in general, we never have $\\boxname(\\chi\\_i)=\\boxname(\\chi\\_j)$ with *j* > *i* + 1: otherwise, by iterating our previous argument, we would obtain $\\boxname(\\chi\\_k)=\\boxname(\\chi\\_i)$ whenever *i* ≤ *k* ≤ *j*, and both *e**k* and *e*ʹ*k* would both be principal boundaries whenever *i* < *k* < *j*.\nIt follows that *p* is acyclic as soon as *P*∘ is. Indeed, if *ξ* is a cycle in *p*:\n* either *ξ* contains an outer tree, and we can apply Lemma [lem:cheminsTaylor] to obtain a cycle in *P*∘;\n* or *ξ* is an inner path, and we proceed inductively in $\\Theta(\\boxname(\\xi))$.\nOur next result is a quantitative version of this property: not only there is no cycle in *p* but the length of paths in *p* is bounded by a function of *P*∘ (whereas the size of *p* is obviously not bounded in general).\n[theorem:length] If $p\\in\\taylor(P^\\circ)$ and $\\xi\\in\\paths(p)$ then $\\ln(\\xi)\\le 2^{\\depth(P^\\circ)}\\size(P^\\circ)$.\nThe proof is by induction on $\\depth(P^\\circ)$.\nFirst assume that *ξ* = *ξ*0(*e*1)*χ*1(*e*ʹ1)*ξ*1⋯(*e**n*)*χ**n*(*e*ʹ*n*)*ξ**n* where each *ξ**i* is an outer path, and each (*e**i*)*χ**i*(*e*ʹ*i*) is a box path. Write $(b\\_i,j\\_i)=\\boxcopy(\\chi\\_i)$: by applying the induction hypothesis to $\\chi\\_i\\in\\paths(p^{b\\_i}\\_{j\\_i})$, we obtain $\\ln(\\chi\\_i)\\le 2^{\\depth(\\Theta(b\\_i))}\\size(\\Theta(b\\_i))$. Moreover observe that $2n+\\sum\\_{i=0}^n \\ln(\\untaylor{\\xi\\_i})\\le \\ln(\\untaylor{\\xi})\\le\\tlsize(P^\\circ)$. By Lemma [lem:outer], it follows that $2n+\\sum\\_{i=0}^n \\ln(\\xi\\_i)\\le \\tlsize(P^\\circ)$. We obtain: \n$$\\ln(\\xi)\n = 2n+\\sum\\_{i=0}^n\\ln(\\xi\\_i)+\\sum\\_{i=1}^{n}\\ln(\\chi\\_i)\n \\le \\tlsize(P^\\circ)\n + \\sum\\_{i=1}^n 2^{\\depth(\\Theta(b\\_i))}\\size(\\Theta(b\\_i)).$$ \n By Lemma [lem:cheminsTaylor], each $b\\in\\boxes\\_\\Theta$ occurs at most twice in the sequence $(b\\_1,\\dotsc,b\\_n)$, hence we obtain: \n$$\\ln(\\xi) \n \\le \\tlsize(P^\\circ)\n + 2\\sum\\_{b\\in\\boxes\\_\\Theta} 2^{\\depth(\\Theta(b))}\\size(\\Theta(b)).$$ \n hence \n$$\\ln(\\xi) \n \\le 2^{\\depth(P^\\circ)} \\big(\\tlsize(P^\\circ)\n + \\sum\\_{b\\in\\boxes\\_\\Theta} \\size(\\Theta(b))\\big).$$ \n since $\\depth(\\Theta(b))<\\depth(P^\\circ)$ for each $b\\in\\boxes\\_\\Theta$. We conclude recalling that $\\size(P^\\circ)=\\tlsize(P^\\circ)+\\sum\\_{b\\in\\boxes\\_\\Theta}\\size(\\Theta(b))$.\nThe other possible cases are those of paths *χ*0(*e*ʹ0)*ξ*, *ξ*(*e**n* + 1)*χ**n* or *χ*0(*e*ʹ0)*ξ*(*e**n* + 1)*χ**n* where *ξ* is as above *e*ʹ0 and *e**n* + 1 are boundaries and *χ*0 and *χ**n* are inner paths. Reasonning as in the proof of Lemma [lem:cheminsTaylor], we also obtain that each $b\\in\\boxes\\_\\Theta$ occurs at most twice in the sequence, e.g., $(b\\_0,\\dotsc,b\\_{n+1})$, and then the proof follows similarly.\nIn particular, we obtain $\\ln(p)\\le 2^{\\depth(P^\\circ)}\\size(P^\\circ)$, In the following lemma, we show that our measure on jumps in the Taylor expansion of *P*∘ is also entirely determined by *P*∘.\n[QITaylor] If $p\\in\\taylor(P^\\circ)$ then $\\jd(p)\\leq\\size(P^\\circ)$.\nWe show that if $t\\in\\ST(p)$ then $\\jd(t)\\le\\size(P^\\circ)$. The proof is, again, by induction on $\\depth(P^\\circ)$. If *t* is inner with $\\boxcopy(t)=(b,j)$, then we conclude directly by applying the induction hypothesis to *p**j**b* and Θ(*b*): indeed in this case, $\\jump^{-1}\\_p(t)=\\jump^{-1}\\_{p^b\\_j}(t)$, and $\\size(\\Theta(b))\\le\\size(P^\\circ)$.\nSo we can assume that *t* is outer. In this case, observe from Definition [defTaylor] that if $\\jump\\_p(\\w)=t$ then $\\w=\\bsubs T{\\vec r}$ for some $T\\in\\ST(P^\\circ)$. It follows that $\\card{\\jump\\_p^{-1}(t)}\\le\\card{\\ST(P^\\circ)}\\le\\size(P^\\circ)$.\nCut elimination and Taylor expansion\n------------------------------------\nIn resource nets, the elimination of the cut \n$$\\cut{\\wn(t\\_1,\\dots,t\\_n)}{\\oc(s\\_1,\\dots,s\\_m)}$$ \n yields the finite sum \n$$\\sum\\_{\\sigma:\\{1,\\dotsc,n\\}\\stackrel\\sim\\to\\{1,\\dotsc,m\\}}\\cut{t\\_1}{s\\_{\\sigma(1)}},\\dots,\\cut{t\\_n}{s\\_{\\sigma(n)}}.$$ \n It turns out that the results of Sections [section:size] to [section:general] apply directly to resource nets: setting \n$$\\cut{\\wn(t\\_1,\\dots,t\\_n)}{\\oc(s\\_1,\\dots,s\\_n)}\\to\n\\cut{t\\_1}{s\\_{\\sigma(1)}},\\dots,\\cut{t\\_n}{s\\_{\\sigma(n)}}$$ \n for each permutation *σ*, we obtain an instance of multiplicative reduction, as the order of premises is irrelevant from a combinatorial point of view — this is all the more obvious because no typing constraint was involved in our argument. In other words, Corollary [corollary:antireducts] also applies to the parallel reduction of resource nets. With Theorem [theorem:length] and Lemma [QITaylor] we obtain:\n[corollary:taylor] If *q* is a resource net and *P* is an MELL net and $k\\in\\N$, $\\{p\\in\\taylor(P)\\tq p\\toto^k q\\}$ is finite.\nAs for Corollaries [corollary:antireducts:onestep] and [corollary:antireducts], this holds only up to *α*-equivalence. And, again, it should be read keeping in mind that jumps are only an additional control structure on top of the underlying net. Indeed, if *P* is a bare MELL net (*i.e.* an MELL net without a jump function) then we can define $\\taylor(P)$ as a set of bare resource nets. Then, given $k\\in\\N$, a bare resource net *q*, a bare MELL net *P*, and a jump function $\\jump$ such that $(P,\\jump)$ acyclic, there are finitely many bare resource nets $p\\in\\taylor(P)$ such that $p\\toto^k q$: it suffices to construct $\\jump\\_p$ from $\\jump$.\nBeware that Corollary [corollary:taylor] depends on the acyclicity of the original MELL net. The following example shows how a cyclic net can induce infinite sets of antireducts.\n[ex:cycle2] Let $P=(\\Theta; \\cut{\\wn(a\\_1^b)}{a\\_0^b};;\\jump)$ with $\\Theta(b)=(\\Theta';;x,\\bar x;;\\jump)$ where the domain of Θʹ, $\\jump$ and $\\jump'$ is empty. Then, by definition, \n$$\\taylor(P)=\\{p=(\\cut{\\cw}{\\w};)\\}\\cup\\{ p\\_n=(\\cut{\\wn(x\\_0,\\dotsc,x\\_n)}{\\oc(\\bar x\\_0,\\dotsc,\\bar x\\_n)};)\\tq n\\in\\N\\}$$ \n where $\\jump\\_{p}(\\cw)=\\w$. Then, for each $n\\in\\N$, we have \n$$p\\_n\\to (\\cut{x\\_0}{\\bar x\\_{\\sigma(0)}},\\dotsc,\\cut{x\\_n}{\\bar x\\_{\\sigma(n)}};)$$ \n for each permutation *σ* of $\\{0,\\dotsc,n\\}$. In particular, if we set $\\sigma(i)=i+1\\mod(n+1)$, then we obtain $p\\_n\\to q\\_n=(\\cut{x\\_0}{\\bar x\\_1},\\dotsc,\\cut{x\\_n}{\\bar x\\_0};)\\totoa(\\cut{x\\_0}{\\bar x\\_0};)$, and it follows that $\\{q\\in\\taylor(P)\\mid q\\toto^2 (\\cut{x\\_0}{\\bar x\\_0};)\\}$ is infinite. This situation is illustrated in Figure [fig:cycle2].\n(a)at(0,.5)ax; (c)at((*a*) + (0, − 1.5))cut; (b)at((*a*) + (0.5, − 0.5))$\\oc$; (w)at((*a*) + ( − 0.5, − 1))$\\wn$; (in) at (b);\n(a)node*x* ; (a)node$\\ \\bar x$;\n(a)to[out=west,in=north](w|-in); (w|-b)–(w); (a)to[out=east,in=north](b); (w)to[out=south,in=west](c); (b)to[out=south,in=east](c); (b)–++(-1.5,0)–++(0,1)–++(1.9,0)–++(0,-1)–(b);\n(P) at (0,-2.5) *P*;\n(pn) at ((*P*) + (4.2, 0)) $p\\_n\\in\\taylor(P)$;\n(c)at((*p**n*) + (0, .8))cut; (w)at((*c*) + ( − 1.25, 1))$\\wn$; (b)at((*c*) + (1.25, 1))$\\oc$; (a1)at((0, 3) + 1/2 \\* ((w)+(b)))ax; (an)at((*a*1) + (0, − 1.3))ax;\n(a1)node*x*0 ; (a1)node$\\ \\bar x\\_0$; (an)node*x**n* ; (an)node$\\ \\bar x\\_n$;\n(a1)to[out=west,in=north west](w); (an)to[out=west,in=north east](w); (a1)to[out=east,in=north west](b); (an)to[out=east,in=north east](b);\n(w)to[out=south,in=west](c); (b)to[out=south,in=east](c);\n((*w*) + (0, 0.5))node…; ((*b*) + (0, 0.5))node…;\n(1/2 \\* ((a1)+(an))) node$\\rvdots$;\n(to) at ((*p**n*) + (2.5, 3)) → ;\n(a1)at((*p**n*) + (5, 5))ax; (an)at((*a*1) + (0, − 1.3))ax; (1/2 \\* ((a1)+(an))) node$\\rvdots$;\n(a1)node*x*0 ; (a1)node$\\ \\bar x\\_0$; (an)node*x**n* ; (an)node$\\ \\bar x\\_n$;\n(c1)at((*a**n*) + ( − 0.8, − 1.4))cut; (cn)at((*c*1) + (0, − 1.4))cut; (1/2 \\* ((c1)+(cn))) node$\\rvdots$;\n(a1)to[out=west,in=west](c1); (an)to[out=west,in=west](cn); (c1)node$\\ \\bar x\\_{\\sigma(0)}$; (cn)node$\\ \\bar x\\_{\\sigma(n)}$;\n((*c*1) + (1.2, 0.2))–++(0.7,0) –((*c**n*) + (1.9, − 0.2))–++(-0.7,0)–cycle; (c1)–++(1.2,0); (cn)–++(1.2,0);\n((*c**n*) + (1.5, 0.6))node*σ*;\n((*c*1) + (1.9, 0))to[out=east,in=east](a1); ((*c**n*) + (1.9, 0))to[out=east,in=east](an);\n(a)at((*a**n*) + (4, 0))ax; (c)at((*a*) + (0, − 1.3))cut; (a)to[out=west,in=west](c); (a)to[out=east,in=east](c);\nat ((*t**o*) + (5, 0)) $\\totoa$;\n(a)node*x*0 ; (a)node$\\ \\bar x\\_0$;\n[fig:cycle2]\nConclusion\n==========\nRecall that our original motivation was the definition of a reduction relation on infinite linear combinations of resource nets, simulating cut elimination in MELL through Taylor expansion. We claim that a suitable notion is as follows:\nWrite ∑*i* ∈ *I**a**i**p**i* ⇒ ∑*i* ∈ *I**a**i**q**i* as soon as:\n* each *p**i* is a resource net and each *q**i* is a finite sum of resource nets such that $p\\_i\\toto q\\_i$;\n* for any resource net *p*, $\\{i\\in I\\tq p\\_i=p\\}$ is finite;\n* for any resource net *q*, $\\{i\\in I\\tq q\\text{ is a summand of }q\\_i\\}$ is finite.\nIn particular, if ∑*i* ∈ *I**a**i**p**i* is a Taylor expansion, then Corollary [corollary:taylor] ensures that the last condition of the definition of ⇒ is automatically valid. The details of the simulation in a quantitative setting remain to be worked out, but the main stumbling block is now over: the necessary equations on coefficients are well established, as they have been extensively studied in the various denotational models; it only remained to be able to form the associated sums directly in the syntax.\nAnother incentive to publish our results is the *normalization-by-evaluation* programme that we develop with Guerrieri, Pellissier and Tortora de Falco . This approach is restricted to connected MELL proof nets, *i.e.* MELL proof nets without weakening, and whose switching graphs are not only acyclic but also connected:[13](#fn13)\n* in this setting, a net *P* is entirely determined by the point of order 2 of its Taylor expansion, *i.e.* the unique resource net $p\\in\\taylor(P)$ with binary cocontractions only ;\n* moreover, given two cut-free *η*-expanded nets *Q* and *R*, both the size of the normal form of a cut between *Q* and *R* and the number of cut elimination steps necessary to reach it can be bounded by a function of the relational semantics of *Q* and *R* ;\n* from this data, we obtain a bound on the size of the point *p*0 of order 2 of the normal form of the cut, as well as a bound on the number of parallel cut elimination steps necessary to obtain *p*0 from its antecedent *p* in the Taylor expansion of the cut.\nOur results in the present paper then provide a bound on the size of *p*: to find *p* it is then sufficient to compute the relational semantics of all the elements of the Taylor expansion of the cut whose size does not exceed this bound, and to check which one gives a semantics of order 2; then we can compute *p*0 as the normal form of *p*, and this is sufficient to determine the normal form of the cut.\nThe restriction to connected nets is necessary to apply the injectivity result of Guerrieri, Pellissier and Tortora de Falco , based on a fixed order of Taylor expansion. The injectivity of Taylor expansion and thus of the relational semantics of full MELL has been proved by de Carvalho : to determine *P* from $\\taylor(P)$, this result relies on a *k*-heterogeneous expansion of *P*, *i.e.* an expansion of *P* for which the number of copies of each box is a power of *k*, and those degrees of expansion are chosen pairwise distinct. For the result to apply, the value of the parameter *k* must be sufficiently large: such a *k* may be computed from the linear expansion of *P*, obtained by taking exactly one copy of each box; but the degrees of expansion of boxes cannot be bounded in advance, and it is thus not clear if the above normalization-by-evaluation procedure could be adapted in this setting.\nLet us conclude with a remark about a possible adaptation of our results to a (maybe) more standard representation of nets, including separate derelictions and coderelictions, with a finer grained cut elimination procedure. This introduces additional complexity in the formalism but it essentially requires no new concept or technique: the difficulty in parallel reduction is to control the chains of cuts to be simultaneously eliminated, and decomposing cut elimination into finer reduction steps can only decrease the length of such chains. On the other hand, in that setting, it is well known that cut elimination alone is not enough to capture the *β*-reduction of *λ*-calculus, and it must be extended with additional rewriting rules accounting for structural identities (e.g., associativity and commutativity of contraction). The details of the Taylor expansion analysis of cut elimination up to these identities are worked out in the PhD thesis of the first author [14](#fn14), including the treatment of coefficients as mentioned above.\nAcknowledgements\n----------------\nWe would like to thank the anonymous referee whose keen remarks have allowed us to clarify several key aspects of our contributions.\nThis work also owes much to the friendly and stimulating environment provided by the International Research Network on Linear Logic[15](#fn15) between the French CNRS and the Italian INDAM: the first steps were actually taken on the occasion of the very first workshop of the network at Roma Tre in December 2015, dedicated to *New trends in linear logic proof-nets*; and successive versions were presented and discussed on various occasions organized by the IRN-LL.\n---\n1. The treatment of weakenings is indeed the main novelty of the present extended version over our conference paper.[↩](#fnref1)\n2. In other words, we adhere to a version of linear logic proof nets and resource nets which is sometimes called *nouvelle syntaxe*, although it dates back to Regnier’s PhD thesis. For the linear logic *connoisseur*, this is already apparent in Figure [fig:taylor]. See also the discussion in our conclusion (Section [section:conclusion]).[↩](#fnref2)\n3. Note that this situation differs slightly from the case of interaction nets , where explicit axioms and cuts links are missing and there is no top-down orientation *a priori*. Term syntaxes have been proposed for those but the correspondence is less immediate: it must be restricted to deadlock-free interaction nets and, in addition to *α*-equivalence, one must introduce some mechanism to deal with implicit axiom-cut elimination in the application of reduction rules.[↩](#fnref3)\n4. Since the cuts of a net are given as a family rather than a sequence, the order in which we write cuts in this definition is not relevant: despite our abusive notation, the reduced cut need not be the first in the enumeration, because this enumeration is not fixed.[↩](#fnref4)\n5. We arbitrarily redirect the jumps to the first subtree to simplify the presentation, but we could equivalently have set $\\jump\\_{p'}(\\w)$ to be any of the immediate subtrees of $\\jump\\_p(\\w)$, non deterministically: in fact, this slight generalization is necessary to deal with cut elimination in resource nets.\nOther strategies for choosing the destination of a jump exist in the literature: for instance, one may be tempted to systematically redirect jumps to atoms, as it is done by Tortora de Falco . But this kind of transformation is not local and it would certainly complicate our arguments.[↩](#fnref5)\n6. Of course, the converse does not hold: for instance the reductions $(\\cut{\\parr(x,\\bar x)}{\\otimes(y,z)};\\bar y,\\bar z)\n\\totom\n(\\cut x y,\\cut {\\bar x} z;\\bar y,\\bar z)\n\\totoa\n(\\cut y z;\\bar y,\\bar z)\n$ cannot be performed in a single step, as the cut $\\cut xy$ was newly created.[↩](#fnref6)\n7. In standard terminology of graph theory, an *I*-path in *p* is a trail in the switching graph induced by *p* and *I*.[↩](#fnref7)\n8. We do not detail the proof as it is quite standard. We will moreover generalize this technique to all paths (and not only cycles) in the next section.[↩](#fnref8)\n9. This is due to the fact that we distinguish between strict connectives and their nullary versions, that are subject to evanescent reductions.[↩](#fnref9)\n10. So, formally, this construction should be parametrized by suitable injections $\\{a^b\\_0\\in\\ST(S^\\circ)\\}\\to \\coweaks$ and $\\{\\wn(T^\\circ\\_1,\\dotsc,T^\\circ\\_n)\\in \\ST(S^\\circ)\\}\\to \\weaks$ to ensure this linearity constraint. We keep this implicit in the following, but will rely on the fact that, given $t\\in\\ST(\\rsubs {S^\\circ}{\\vec r})$, one can recover unambiguously one of the following: either $T\\in\\ST(S^\\circ)$ such that $t=\\bsubs {T}{\\vec r}$; or *b* and *j* such that $t\\in\\ST(\\vec s^b\\_j)$.[↩](#fnref10)\n11. More extensive presentations of the Taylor expansion of MELL nets exist in the literature, in various styles. Our only purpose here is to introduce sufficient notations to present our analysis of the jump degree and the length of paths in $\\taylor(P)$ w.r.t. the size of *P*.[↩](#fnref11)\n12. Observe that there might be several possible choices for *T**i*∘ so $\\untaylor I$ is not uniquely defined in this manner: our following constructions thus depend on the choices we make for $\\untaylor I$.[↩](#fnref12)\n13. These are sufficiently expressive to simulate the *λ**I*-calculus, which is Turing-complete.[↩](#fnref13)\n14. In French.[↩](#fnref14)\n15. <http://www.linear-logic.org/>[↩](#fnref15)\n"},"data_id":{"kind":"string","value":"arxiv_0185578"}}},{"rowIdx":46399,"cells":{"content":{"kind":"string","value":"5]; at (-0.25-3.5,-0.75)[circle,draw=black,fill=black,scale=0.5]; at (-0.75-3.5,-0.75)[circle,draw=black,fill=black,scale=0.5]; at (0.25-3.5,1.25)[circle,draw=black,fill=black,scale=0.5]; at (-0.25-3.5,1.25)[circle,draw=black,fill=black,scale=0.5]; at (0.25-3.5,-1.25)[circle,draw=black,fill=black,scale=0.5]; at (-0.25-3.5,-1.25)[circle,draw=black,fill=black,scale=0.5];\nat (0.45-3.5,0.4); at (0.95-3.5,0.4); at (1.45-3.5,0.4); at (-0.05-3.5,0.4); at (-0.55-3.5,0.4); at (-1.05-3.5,0.4); at (0.45-3.5,-0.1); at (0.95-3.5,-0.1); at (1.45-3.5,-0.1); at (-0.05-3.5,-0.1); at (-0.55-3.5,-0.1); at (-1.05-3.5,-0.1); at (0.45-3.5,0.9); at (0.95-3.5,0.9); at (-0.05-3.5,0.9); at (-0.55-3.5,0.9); at (0.45-3.5,-0.6); at (0.95-3.5,-0.6); at (-0.05-3.5,-0.6); at (-0.55-3.5,-0.6); at (0.45-3.5,1.4); at (-0.05-3.5,1.4); at (0.45-3.5,-1.1); at (-0.05-3.5,-1.1);\n(-0.5-7,1) – (0.5-7,1); (-1-7,0.5) – (1-7,0.5); (-1-7,0) – (1-7,0); (-1-7,-0.5) – (1-7,-0.5); (-0.5-7,-1) – (0.5-7,-1); (1-7,-0.5) – (1-7,0.5); (0.5-7,-1) – (0.5-7,1); (0-7,-1) – (0-7,1); (-0.5-7,-1) – (-0.5-7,1); (-1-7,-0.5) – (-1-7,0.5);\nat (-0.5-7,1)[circle,draw=black,fill=black,scale=0.5]; at (0-7,1)[circle,draw=black,fill=black,scale=0.5]; at (0.5-7,1)[circle,draw=black,fill=black,scale=0.5]; at (-0.5-7,0.5)[circle,draw=black,fill=black,scale=0.5]; at (-1-7,0.5)[circle,draw=black,fill=black,scale=0.5]; at (0-7,0.5)[circle,draw=black,fill=black,scale=0.5]; at (0.5-7,0.5)[circle,draw=black,fill=black,scale=0.5]; at (1-7,0.5)[circle,draw=black,fill=black,scale=0.5]; at (-0.5+7,0)[circle,draw=black,fill=black,scale=0.5]; at (-1-7,0)[circle,draw=black,fill=black,scale=0.5]; at (0-7,0)[circle,draw=black,fill=black,scale=0.5]; at (0.5-7,0)[circle,draw=black,fill=black,scale=0.5]; at (1-7,0)[circle,draw=black,fill=black,scale=0.5]; at (-0.5-7,-0.5)[circle,draw=black,fill=black,scale=0.5]; at (-1-7,-0.5)[circle,draw=black,fill=black,scale=0.5]; at (0-7,-0.5)[circle,draw=black,fill=black,scale=0.5]; at (0.5-7,-0.5)[circle,draw=black,fill=black,scale=0.5]; at (1-7,-0.5)[circle,draw=black,fill=black,scale=0.5]; at (-0.5-7,-1)[circle,draw=black,fill=black,scale=0.5]; at (0-7,-1)[circle,draw=black,fill=black,scale=0.5]; at (0.5-7,-1)[circle,draw=black,fill=black,scale=0.5];\nat (0.2-7,0.15); at (0.7-7,0.15); at (1.2-7,0.15); at (-0.3-7,0.15); at (-0.8-7,0.15); at (0.2-7,0.65); at (0.7-7,0.65); at (1.2-7,0.65); at (-0.3-7,0.65); at (-0.8-7,0.65); at (0.2-7,-0.35); at (0.7-7,-0.35); at (1.2-7,-0.35); at (-0.3-7,-0.35); at (-0.8-7,-0.35); at (0.2-7,1.15); at (0.7-7,1.15); at (-0.3-7,1.15); at (0.2-7,-0.85); at (0.7-7,-0.85); at (-0.3-7,-0.85);\n(-0.75-10.5,0.75) – (0.75-10.5,0.75); (-0.75-10.5,0.25) – (0.75-10.5,0.25); (-0.75-10.5,-0.25) – (0.75-10.5,-0.25); (-0.75-10.5,-0.75) – (0.75-10.5,-0.75); (-0.75-10.5,-0.75) – (-0.75-10.5,0.75); (-0.25-10.5,-0.75) – (-0.25-10.5,0.75); (0.25-10.5,-0.75) – (0.25-10.5,0.75); (0.75-10.5,-0.75) – (0.75-10.5,0.75);\nat (-0.75-10.5,0.75)[circle,draw=black,fill=black,scale=0.5]; at (-0.75-10.5,0.25)[circle,draw=black,fill=black,scale=0.5]; at (-0.75-10.5,-0.25)[circle,draw=black,fill=black,scale=0.5]; at (-0.75-10.5,-0.75)[circle,draw=black,fill=black,scale=0.5]; at (-0.25-10.5,0.75)[circle,draw=black,fill=black,scale=0.5]; at (-0.25-10.5,0.25)[circle,draw=black,fill=black,scale=0.5]; at (-0.25-10.5,-0.25)[circle,draw=black,fill=black,scale=0.5]; at (-0.25-10.5,-0.75)[circle,draw=black,fill=black,scale=0.5]; at (0.75-10.5,0.75)[circle,draw=black,fill=black,scale=0.5]; at (0.75-10.5,0.25)[circle,draw=black,fill=black,scale=0.5]; at (0.75-10.5,-0.25)[circle,draw=black,fill=black,scale=0.5]; at (0.75-10.5,-0.75)[circle,draw=black,fill=black,scale=0.5]; at (0.25-10.5,0.75)[circle,draw=black,fill=black,scale=0.5]; at (0.25-10.5,0.25)[circle,draw=black,fill=black,scale=0.5]; at (0.25-10.5,-0.25)[circle,draw=black,fill=black,scale=0.5]; at (0.25-10.5,-0.75)[circle,draw=black,fill=black,scale=0.5];\nat (0.45-10.5,0.4); at (0.95-10.5,0.4); at (-0.05-10.5,0.4); at (-0.55-10.5,0.4); at (0.45-10.5,0.9); at (0.95-10.5,0.9); at (-0.05-10.5,0.9); at (-0.55-10.5,0.9); at (0.45-10.5,-0.1); at (0.95-10.5,-0.1); at (-0.05-10.5,-0.1); at (-0.55-10.5,-0.1); at (0.45-10.5,-0.6); at (0.95-10.5,-0.6); at (-0.05-10.5,-0.6); at (-0.55-10.5,-0.6);\n[scale=1.15] (-1.25+3.5,0.25) – (1.25+3.5,0.25); (-0.75+3.5,0.75) – (0.75+3.5,0.75); (-0.25+3.5,1.25) – (0.25+3.5,1.25); (-1.25+3.5,-0.25) – (1.25+3.5,-0.25); (-0.75+3.5,-0.75) – (0.75+3.5,-0.75); (-0.25+3.5,-1.25) – (0.25+3.5,-1.25);\n(0.25+3.5,-1.25) – (0.25+3.5,1.25); (0.75+3.5,-0.75) – (0.75+3.5,0.75); (1.25+3.5,-0.25) – (1.25+3.5,0.25); (-0.25+3.5,-1.25) – (-0.25+3.5,1.25); (-0.75+3.5,-0.75) – (-0.75+3.5,0.75); (-1.25+3.5,-0.25) – (-1.25+3.5,0.25);\nat (0.25+3.5,0.25)[circle,draw=black,fill=black,scale=0.5]; at (0.75+3.5,0.25)[circle,draw=black,fill=black,scale=0.5]; at (1.25+3.5,0.25)[circle,draw=black,fill=black,scale=0.5]; at (-0.25+3.5,0.25)[circle,draw=black,fill=black,scale=0.5]; at (-0.75+3.5,0.25)[circle,draw=black,fill=black,scale=0.5]; at (-1.25+3.5,0.25)[circle,draw=black,fill=black,scale=0.5]; at (0.25+3.5,-0.25)[circle,draw=black,fill=black,scale=0.5]; at (0.75+3.5,-0.25)[circle,draw=black,fill=black,scale=0.5]; at (1.25+3.5,-0.25)[circle,draw=black,fill=black,scale=0.5]; at (-0.25+3.5,-0.25)[circle,draw=black,fill=black,scale=0.5]; at (-0.75+3.5,-0.25)[circle,draw=black,fill=black,scale=0.5]; at (-1.25+3.5,-0.25)[circle,draw=black,fill=black,scale=0.5]; at (0.25+3.5,0.75)[circle,draw=black,fill=black,scale=0.5]; at (0.75+3.5,0.75)[circle,draw=black,fill=black,scale=0.5]; at (-0.25+3.5,0.75)[circle,draw=black,fill=black,scale=0.5]; at (-0.75+3.5,0.75)[circle,draw=black,fill=black,scale=0.5]; at (0.25+3.5,-0.75)[circle,draw=black,fill=black,scale=0.5]; at (0.75+3.5,-0.75)[circle,draw=black,fill=black,scale=0.5]; at (-0.25+3.5,-0.75)[circle,draw=black,fill=black,scale=0.5]; at (-0.75+3.5,-0.75)[circle,draw=black,fill=black,scale=0.5]; at (0.25+3.5,1.25)[circle,draw=black,fill=black,scale=0.5]; at (-0.25+3.5,1.25)[circle,draw=black,fill=black,scale=0.5]; at (0.25+3.5,-1.25)[circle,draw=black,fill=black,scale=0.5]; at (-0.25+3.5,-1.25)[circle,draw=black,fill=black,scale=0.5];\nat (0.45+3.5,0.4); at (0.95+3.5,0.4); at (1.45+3.5,0.4); at (-0.05+3.5,0.4); at (-0.55+3.5,0.4); at (-1.05+3.5,0.4); at (0.45+3.5,-0.1); at (0.95+3.5,-0.1); at (1.45+3.5,-0.1); at (-0.05+3.5,-0.1); at (-0.55+3.5,-0.1); at (-1.05+3.5,-0.1); at (0.45+3.5,0.9); at (0.95+3.5,0.9); at (-0.05+3.5,0.9); at (-0.55+3.5,0.9); at (0.45+3.5,-0.6); at (0.95+3.5,-0.6); at (-0.05+3.5,-0.6); at (-0.55+3.5,-0.6); at (0.45+3.5,1.4); at (-0.05+3.5,1.4); at (0.45+3.5,-1.1); at (-0.05+3.5,-1.1);\n(-0.5+7,1) – (0.5+7,1); (-1+7,0.5) – (1+7,0.5); (-1+7,0) – (1+7,0); (-1+7,-0.5) – (1+7,-0.5); (-0.5+7,-1) – (0.5+7,-1); (1+7,-0.5) – (1+7,0.5); (0.5+7,-1) – (0.5+7,1); (0+7,-1) – (0+7,1); (-0.5+7,-1) – (-0.5+7,1); (-1+7,-0.5) – (-1+7,0.5);\nat (-0.5+7,1)[circle,draw=black,fill=black,scale=0.5]; at (0+7,1)[circle,draw=black,fill=black,scale=0.5]; at (0.5+7,1)[circle,draw=black,fill=black,scale=0.5]; at (-0.5+7,0.5)[circle,draw=black,fill=black,scale=0.5]; at (-1+7,0.5)[circle,draw=black,fill=black,scale=0.5]; at (0+7,0.5)[circle,draw=black,fill=black,scale=0.5]; at (0.5+7,0.5)[circle,draw=black,fill=black,scale=0.5]; at (1+7,0.5)[circle,draw=black,fill=black,scale=0.5]; at (-0.5+7,0)[circle,draw=black,fill=black,scale=0.5]; at (-1+7,0)[circle,draw=black,fill=black,scale=0.5]; at (0+7,0)[circle,draw=black,fill=black,scale=0.5]; at (0.5+7,0)[circle,draw=black,fill=black,scale=0.5]; at (1+7,0)[circle,draw=black,fill=black,scale=0.5]; at (-0.5+7,-0.5)[circle,draw=black,fill=black,scale=0.5]; at (-1+7,-0.5)[circle,draw=black,fill=black,scale=0.5]; at (0+7,-0.5)[circle,draw=black,fill=black,scale=0.5]; at (0.5+7,-0.5)[circle,draw=black,fill=black,scale=0.5]; at (1+7,-0.5)[circle,draw=black,fill=black,scale=0.5]; at (-0.5+7,-1)[circle,draw=black,fill=black,scale=0.5]; at (0+7,-1)[circle,draw=black,fill=black,scale=0.5]; at (0.5+7,-1)[circle,draw=black,fill=black,scale=0.5];\nat (0.2+7,0.15); at (0.7+7,0.15); at (1.2+7,0.15); at (-0.3+7,0.15); at (-0.8+7,0.15); at (0.2+7,0.65); at (0.7+7,0.65); at (1.2+7,0.65); at (-0.3+7,0.65); at (-0.8+7,0.65); at (0.2+7,-0.35); at (0.7+7,-0.35); at (1.2+7,-0.35); at (-0.3+7,-0.35); at (-0.8+7,-0.35); at (0.2+7,1.15); at (0.7+7,1.15); at (-0.3+7,1.15); at (0.2+7,-0.85); at (0.7+7,-0.85); at (-0.3+7,-0.85);\n(-0.75+10.5,0.75) – (0.75+10.5,0.75); (-0.75+10.5,0.25) – (0.75+10.5,0.25); (-0.75+10.5,-0.25) – (0.75+10.5,-0.25); (-0.75+10.5,-0.75) – (0.75+10.5,-0.75); (-0.75+10.5,-0.75) – (-0.75+10.5,0.75); (-0.25+10.5,-0.75) – (-0.25+10.5,0.75); (0.25+10.5,-0.75) – (0.25+10.5,0.75); (0.75+10.5,-0.75) – (0.75+10.5,0.75);\nat (-0.75+10.5,0.75)[circle,draw=black,fill=black,scale=0.5]; at (-0.75+10.5,0.25)[circle,draw=black,fill=black,scale=0.5]; at (-0.75+10.5,-0.25)[circle,draw=black,fill=black,scale=0.5]; at (-0.75+10.5,-0.75)[circle,draw=black,fill=black,scale=0.5]; at (-0.25+10.5,0.75)[circle,draw=black,fill=black,scale=0.5]; at (-0.25+10.5,0.25)[circle,draw=black,fill=black,scale=0.5]; at (-0.25+10.5,-0.25)[circle,draw=black,fill=black,scale=0.5]; at (-0.25+10.5,-0.75)[circle,draw=black,fill=black,scale=0.5]; at (0.75+10.5,0.75)[circle,draw=black,fill=black,scale=0.5]; at (0.75+10.5,0.25)[circle,draw=black,fill=black,scale=0.5]; at (0.75+10.5,-0.25)[circle,draw=black,fill=black,scale=0.5]; at (0.75+10.5,-0.75)[circle,draw=black,fill=black,scale=0.5]; at (0.25+10.5,0.75)[circle,draw=black,fill=black,scale=0.5]; at (0.25+10.5,0.25)[circle,draw=black,fill=black,scale=0.5]; at (0.25+10.5,-0.25)[circle,draw=black,fill=black,scale=0.5]; at (0.25+10.5,-0.75)[circle,draw=black,fill=black,scale=0.5];\nat (0.45+10.5,0.4); at (0.95+10.5,0.4); at (-0.05+10.5,0.4); at (-0.55+10.5,0.4); at (0.45+10.5,0.9); at (0.95+10.5,0.9); at (-0.05+10.5,0.9); at (-0.55+10.5,0.9); at (0.45+10.5,-0.1); at (0.95+10.5,-0.1); at (-0.05+10.5,-0.1); at (-0.55+10.5,-0.1); at (0.45+10.5,-0.6); at (0.95+10.5,-0.6); at (-0.05+10.5,-0.6); at (-0.55+10.5,-0.6);\nat (3.5,-1.75)F*k* + 1; at (7,-1.75)F*k* + 2; at (10.5,-1.75)F*k* + 3;\n[colorpower3]\nFor vertices *u*1, *u*2, *u*3 and *u*4 of F, we denote by *T*(*u*1, *u*2, *u*3, *u*4), the vertex set {*u* ∈ F∣ *u**v* ∈ *E*(F), *v* ∈ {*u*1, *u*2, *u*3, *u*4}}. Note that in the case {*u*1, *u*2, *u*3, *u*4} induces a complete graph in F, *T*(*u*1, *u*2, *u*3, *u*4) induces in F a graph isomorphic to the graph *B*1 from Lemma [le2]. Consequently, in this case, diam(*T*(*u*1, *u*2, *u*3, *u*4)) = 3.\n[balancetonlemme] If there exists a 28-coloring of F3, then the following is true about every vertex (*i*, *j*, *k*):\n1. one vertex among (*i* + 3, *j*, *k* + 2), (*i* + 2.5, *j* + 0.5, *k* + 3) and (*i* + 2.5, *j* − 0.5, *k* + 3) has the same color than (*i*, *j*, *k*);\n2. one vertex among (*i* − 3, *j*, *k* + 2), (*i* − 2.5, *j* + 0.5, *k* + 3) and (*i* − 2.5, *j* − 0.5, *k* + 3) has the same color than (*i*, *j*, *k*);\n3. one vertex among (*i*, *j* − 3, *k* + 2), (*i* + 0.5, *j* − 2.5, *k* + 3) and (*i* − 0.5, *j* − 2.5, *k* + 3) has the same color than (*i*, *j*, *k*);\n4. one vertex among (*i*, *j* − 3, *k* − 2), (*i* + 0.5, *j* − 2.5, *k* − 3) and (*i* − 0.5, *j* − 2.5, *k* − 3) has the same color than (*i*, *j*, *k*).\n[scale=0.9]\n(0-1,2.5/3-0.5) – (2,2.5/3-0.5); (0-0.5,5/3-0.5) – (3-0.5,5/3-0.5); (0.5-0.5,2) – (3,2); (-1,2.5/3-0.5) – (0,2); (0,2.5/3-0.5) – (1,2); (1,2.5/3-0.5) – (2,2); (2,2.5/3-0.5) – (3,2);\n(0+3.5,0) – (2+3.5,0); (0.5+3.5,2.5/3) – (2.5+3.5,2.5/3); (1+3.5,5/3) – (3+3.5,5/3); (1.5+3.5,2.5) – (3.5+3.5,2.5); (0+3.5,0) – (1.5+3.5,2.5); (1+3.5,0) – (2.5+3.5,2.5); (2+3.5,0) – (3.5+3.5,2.5);\n(0-1+8.5,2.5/3-0.5) – (2+8.5,2.5/3-0.5); (0-0.5+8.5,5/3-0.5) – (3-0.5+8.5,5/3-0.5); (0.5-0.5+8.5,2) – (3+8.5,2); (-1+8.5,2.5/3-0.5) – (0+8.5,2); (0+8.5,2.5/3-0.5) – (1+8.5,2); (1+8.5,2.5/3-0.5) – (2+8.5,2); (2+8.5,2.5/3-0.5) – (3+8.5,2);\n(0+12,0) – (2+12,0); (0.5+12,2.5/3) – (2.5+12,2.5/3); (1+12,5/3) – (3+12,5/3); (1.5+12,2.5) – (3.5+12,2.5); (0+12,0) – (1.5+12,2.5); (1+12,0) – (2.5+12,2.5); (2+12,0) – (3.5+12,2.5);\n(0,2.5/3-0.5) – (1,2); (1,2.5/3-0.5) – (2,2); (1-1,2.5/3-0.5) – (1,2.5/3-0.5); (1-0.5,5/3-0.5) – (2-0.5,5/3-0.5); (1.5-0.5,2) – (2,2);\n(0.5+3.5,2.5/3) – (1+3.5,5/3); (1+3.5,0) – (2.5+3.5,2.5); (2.5+3.5,2.5/3) – (3+3.5,5/3); (1.5/3+3.5,2.5/3) – (2+1.5/3+3.5,2.5/3); (1+3.5,5/3) – (3+3.5,5/3);\n(0+8.5,2.5/3-0.5) – (1+8.5,2); (1+8.5,2.5/3-0.5) – (2+8.5,2); (1-1+8.5,2.5/3-0.5) – (1+8.5,2.5/3-0.5); (-0.5+8.5,5/3-0.5) – (3-0.5+8.5,5/3-0.5); (1.5-0.5+8.5,2) – (2+8.5,2);\n(0.5+12,2.5/3) – (1+12,5/3); (1.5+12,2.5/3) – (2+12,5/3); (2.5+12,2.5/3) – (3+12,5/3); (1.5/3+12,2.5/3) – (2+1.5/3+12,2.5/3); (1+12,5/3) – (3+12,5/3);\nat (0,2.5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (1,2.5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (-0.5,5/3-0.5)[regular polygon, regular polygon sides=4,draw=black,fill=red,scale=0.5]; at (0.5,5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (1.5,5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (1,2)[circle,draw=black,fill=black,scale=0.5]; at (2,2)[circle,draw=black,fill=black,scale=0.5];\nat (2.5+3.5,2.5)[circle,draw=black,fill=black,scale=0.5]; at (3+3.5,5/3)[circle,draw=black,fill=black,scale=0.5]; at (2.5+3.5,2.5/3)[circle,draw=black,fill=black,scale=0.5]; at (3+3.5-1,5/3)[circle,draw=black,fill=black,scale=0.5]; at (2.5+3.5-1,2.5/3)[circle,draw=black,fill=black,scale=0.5]; at (3+3.5-2,5/3)[circle,draw=black,fill=black,scale=0.5]; at (4.5+3.5-4,2.5/3)[circle,draw=black,fill=black,scale=0.5]; at (1+3.5,0)[circle,draw=black,fill=black,scale=0.5]; at (0+8.5,2.5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (1+8.5,2.5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (-0.5+8.5,5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (0.5+8.5,5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (1.5+8.5,5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (2.5+8.5,5/3-0.5)[regular polygon, regular polygon sides=3,draw=black,fill=green,scale=0.35]; at (1+8.5,2)[circle,draw=black,fill=black,scale=0.5]; at (2+8.5,2)[circle,draw=black,fill=black,scale=0.5];\nat (3+12,5/3)[regular polygon, regular polygon sides=3,draw=black,fill=green,scale=0.35]; at (1.5+12,2.5/3)[circle,draw=black,fill=black,scale=0.5]; at (3+12-1,5/3)[circle,draw=black,fill=black,scale=0.5]; at (3.5+12-1,2.5/3)[regular polygon, regular polygon sides=3,draw=black,fill=green,scale=0.35]; at (3+12-2,5/3)[circle,draw=black,fill=black,scale=0.5]; at (4.5+12-4,2.5/3)[circle,draw=black,fill=black,scale=0.5];\nat (0.5,-0.5)F*i*; at (4.5,-0.5)F*i* + 1; at (9,-0.5)F*i* + 2; at (4.5+8.5,-0.5)F*i* + 3;\n[petitfrereveutgrandirtropvite]\n(i) Let *B* = *T*((*i* + 1.5, *j* − 0.5, *k* + 1), (*i* + 1.5, *j* + 0.5, *k* + 1), (*i* + 1, *j*, *k* + 2), (*i* + 2, *j*, *k* + 2)). Let *B*ʹ = *B* ∪ {(*i*, *j*, *k*)}. Since *B* induces in F a graph isomorphic to the graph *B*1 from Lemma [le2], we have ∣*B*∣ = 28. Note that (*i*, *j*, *k*) is at distance at most 3 from every vertex of *B* except (*i* + 3, *j*, *k* + 2), (*i* + 2.5, *j* + 0.5, *k* + 3) and (*i* + 2.5, *j* − 0.5, *k* + 3). Figure [petitfrereveutgrandirtropvite] illustrates the set *B*, the vertex (*i*, *j*, *k*) and the vertices of *B* at distance at most 3 or not from (*i*, *j*, *k*). Since ∣*B*ʹ∣ = 29, in every possible 28-coloring of F3, the vertex (*i*, *j*, *k*) has the same color than a vertex among (*i* + 3, *j*, *k* + 2), (*i* + 2.5, *j* + 0.5, *k* + 3) and (*i* + 2.5, *j* − 0.5, *k* + 3).(ii), (iii) and (iv) By symmetry, the proof is totally analogous to the proof of Case (i) (the set *B* should be chosen differently).\nNote that the lower bound of Proposition [oddbound] (the value 28) corresponds to the size of the largest clique in F3. However, the next result shows that the value of *χ*(F3) is larger than 28.\n*χ*(F3) ≥ 29.\nLet *B* = *T*((0, 0, 0), (0, 1, 0), ( − 0.5, 0.5, 1), (0.5, 0.5, 1)) and let *B*ʹ = *T*((2, 0, 0), (2, 1, 0), (1.5, 0.5, 1), (2.5, 0.5, 1)). Note that ∣*B*∣ = 28 and ∣*B*ʹ∣ = 28 and as in Lemma [balancetonlemme] both *B* and *B*ʹ induce in F a graph isomorphic to the graph *B*1 from Lemma [le2]. Suppose that F3 is 28-colorable. By hypothesis, both *B* and *B*ʹ should contain a vertex of every color. Let *u* = (1, 2, 0), *v*1 = ( − 1, 0, 0) and *v*2 = (3, 0, 0). Moreover, let *X*1 = {( − 0.5, − 0.5, − 1), (2.5, − 0.5, − 1), (0, − 1, 0), (2, − 1, 0)}, *X*2 = ( − 0.5, − 0.5, 1)), (0.5, − 0.5, 1}, *X*3 = {( − 1.5, 0.5, 1)), ( − 1, 0, 2)} and *X*4 = {(3.5, 0.5, 1), (3, 0, 2)}. Figure [ecoutecabb] illustrates the sets *B*, *B*ʹ, *X*1, *X*2, *X*3 and *X*4 and the vertices *u*, *v*1 and *v*2.\nRemark that *u* is at distance at most 3 from every vertex of *B* ∪ *B*ʹ \\ (*X*1 ∪ *X*2 ∪ *X*3 ∪ *X*4 ∪ {*v*1, *v*2}). Figure [ecoutecabb] illustrates the vertices of *B* ∪ *B*ʹ at distance at most 3 of *u*.\n[scale=0.7]\n(0-1,2.5/3-0.5) – (4,2.5/3-0.5); (0-0.5,5/3-0.5) – (5-0.5,5/3-0.5); (0.5-0.5,2) – (5,2); (-1,2.5/3-0.5) – (0,2); (0,2.5/3-0.5) – (1,2); (1,2.5/3-0.5) – (2,2); (2,2.5/3-0.5) – (3,2); (3,2.5/3-0.5) – (4,2); (4,2.5/3-0.5) – (5,2);\n(5.5,0) – (4+5.5,0); (1.5/3+5.5,2.5/3) – (4+1.5/3+5.5,2.5/3); (1+5.5,5/3) – (5+5.5,5/3); (1.5+5.5,2.5) – (1.5+4+5.5,2.5); (0+5.5,0) – (1.5+5.5,2.5); (1+5.5,0) – (2.5+5.5,2.5); (2+5.5,0) – (3.5+5.5,2.5); (3+5.5,0) – (4.5+5.5,2.5); (4+5.5,0) – (5.5+5.5,2.5);\n(0+1.5/3+10.5,2.5/3-0.5) – (5+1.5/3+10.5,2.5/3-0.5); (0+1+10.5,5/3-0.5) – (6+10.5,5/3-0.5); (1.5+10.5,2.5-0.5) – (6.5+10.5,2.5-0.5); (0.5+10.5,2.5/3-0.5) – (1.5+10.5,2); (1.5+10.5,2.5/3-0.5) – (2.5+10.5,2); (2.5+10.5,2.5/3-0.5) – (3.5+10.5,2); (3.5+10.5,2.5/3-0.5) – (4.5+10.5,2); (4.5+10.5,2.5/3-0.5) – (5.5+10.5,2); (5.5+10.5,2.5/3-0.5) – (6.5+10.5,2);\n(0+17.5,0) – (4+17.5,0); (0+1.5/3+17.5,2.5/3) – (4+1.5/3+17.5,2.5/3); (0+1+17.5,5/3) – (5+17.5,5/3); (1.5+17.5,2.5) – (5.5+17.5,2.5); (0+17.5,0) – (1.5+17.5,2.5); (1+17.5,0) – (2.5+17.5,2.5); (2+17.5,0) – (3.5+17.5,2.5); (3+17.5,0) – (4.5+17.5,2.5); (4+17.5,0) – (5.5+17.5,2.5);\n(0,2.5/3-0.5) – (1,2); (1,2.5/3-0.5) – (2,2); (2,2.5/3-0.5) – (3,2); (3,2.5/3-0.5) – (4,2); (1-1,2.5/3-0.5) – (1,2.5/3-0.5); (1-0.5,5/3-0.5) – (2-0.5,5/3-0.5); (1.5-0.5,2) – (2,2); (3-1,2.5/3-0.5) – (3,2.5/3-0.5); (3-0.5,5/3-0.5) – (4-0.5,5/3-0.5); (3.5-0.5,2) – (4,2);\n(0.5+5.5,2.5/3) – (1+5.5,5/3); (1+5.5,0) – (2.5+5.5,2.5); (2.5+5.5,2.5/3) – (3+5.5,5/3); (2.5+5.5,2.5/3) – (3+5.5,5/3); (3+5.5,0) – (4.5+5.5,2.5); (4.5+5.5,2.5/3) – (5+5.5,5/3); (1.5/3+5.5,2.5/3) – (2+1.5/3+5.5,2.5/3); (1+5.5,5/3) – (3+5.5,5/3); (2+1.5/3+5.5,2.5/3) – (4+1.5/3+5.5,2.5/3); (3+5.5,5/3) – (5+5.5,5/3);\n(0+12,2.5/3-0.5) – (1+12,2); (1+12,2.5/3-0.5) – (2+12,2); (2+12,2.5/3-0.5) – (3+12,2); (3+12,2.5/3-0.5) – (4+12,2); (1-1+12,2.5/3-0.5) – (1+12,2.5/3-0.5); (0-0.5+12,5/3-0.5) – (3-0.5+12,5/3-0.5); (1.5-0.5+12,2) – (2+12,2); (3-1+12,2.5/3-0.5) – (3+12,2.5/3-0.5); (2-0.5+12,5/3-0.5) – (5-0.5+12,5/3-0.5); (3.5-0.5+12,2) – (4+12,2);\n(0.5+17.5,2.5/3) – (1+17.5,5/3); (1.5+17.5,2.5/3) – (2+17.5,5/3); (2.5+17.5,2.5/3) – (3+17.5,5/3); (2.5+17.5,2.5/3) – (3+17.5,5/3); (3.5+17.5,2.5/3) – (4+17.5,5/3); (4.5+17.5,2.5/3) – (5+17.5,5/3); (1.5/3+17.5,2.5/3) – (2+1.5/3+17.5,2.5/3); (1+17.5,5/3) – (3+17.5,5/3); (2+1.5/3+17.5,2.5/3) – (4+1.5/3+17.5,2.5/3); (3+17.5,5/3) – (5+17.5,5/3);\nat (0,2.5/3-0.5)[regular polygon, regular polygon sides=3,draw=black,fill=green,scale=0.35]; at (1,2.5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (2,2.5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (3,2.5/3-0.5)[regular polygon, regular polygon sides=3,draw=black,fill=green,scale=0.35]; at (0.5,5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (1.5,5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (2.5,5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (3.5,5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (1,2)[circle,draw=black,fill=black,scale=0.5]; at (2,2)[circle,draw=black,fill=black,scale=0.5]; at (3,2)[circle,draw=black,fill=black,scale=0.5]; at (4,2)[circle,draw=black,fill=black,scale=0.5];\nat (3.5+5.5,2.5)[regular polygon, regular polygon sides=4,draw=black,fill=red,scale=0.5]; at (2.5+5.5,2.5)[circle,draw=black,fill=black,scale=0.5]; at (4.5+5.5,2.5)[circle,draw=black,fill=black,scale=0.5]; at (3+5.5,5/3)[circle,draw=black,fill=black,scale=0.5]; at (2.5+5.5,2.5/3)[circle,draw=black,fill=black,scale=0.5]; at (3+5.5-1,5/3)[circle,draw=black,fill=black,scale=0.5]; at (2.5+5.5-1,2.5/3)[circle,draw=black,fill=black,scale=0.5]; at (3+5.5+1,5/3)[circle,draw=black,fill=black,scale=0.5]; at (2.5+5.5+1,2.5/3)[circle,draw=black,fill=black,scale=0.5]; at (3+5.5-2,5/3)[circle,draw=black,fill=black,scale=0.5]; at (3+5.5+2,5/3)[circle,draw=black,fill=black,scale=0.5]; at (4.5+5.5-4,2.5/3)[regular polygon, regular polygon sides=3,draw=black,fill=green,scale=0.35]; at (4.5+5.5,2.5/3)[regular polygon, regular polygon sides=3,draw=black,fill=green,scale=0.35]; at (1+5.5,0)[regular polygon, regular polygon sides=3,draw=black,fill=green,scale=0.35]; at (3+5.5,0)[regular polygon, regular polygon sides=3,draw=black,fill=green,scale=0.35];\nat (1.5+11-0.5,2.5/3-0.5)[regular polygon, regular polygon sides=3,draw=black,fill=green,scale=0.35]; at (2.5+11-0.5,2.5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (3.5+11-0.5,2.5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (4.5+11-0.5,2.5/3-0.5)[regular polygon, regular polygon sides=3,draw=black,fill=green,scale=0.35];\nat (1+11-0.5,5/3-0.5)[regular polygon, regular polygon sides=3,draw=black,fill=green,scale=0.35]; at (2+11-0.5,5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (3+11-0.5,5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (4+11-0.5,5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (5+11-0.5,5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (6+11-0.5,5/3-0.5)[regular polygon, regular polygon sides=3,draw=black,fill=green,scale=0.35]; at (2.5+11-0.5,2)[circle,draw=black,fill=black,scale=0.5]; at (3.5+11-0.5,2)[circle,draw=black,fill=black,scale=0.5]; at (4.5+11-0.5,2)[circle,draw=black,fill=black,scale=0.5]; at (5.5+11-0.5,2)[circle,draw=black,fill=black,scale=0.5];\nat (3+17.5,5/3)[circle,draw=black,fill=black,scale=0.5]; at (2.5+17.5,2.5/3)[circle,draw=black,fill=black,scale=0.5]; at (3+17.5-1,5/3)[circle,draw=black,fill=black,scale=0.5]; at (2.5+17.5-1,2.5/3)[circle,draw=black,fill=black,scale=0.5]; at (3+17.5+1,5/3)[circle,draw=black,fill=black,scale=0.5]; at (2.5+17.5+1,2.5/3)[circle,draw=black,fill=black,scale=0.5]; at (3+17.5-2,5/3)[circle,draw=black,fill=black,scale=0.5]; at (3+17.5+2,5/3)[circle,draw=black,fill=black,scale=0.5]; at (4.5+17.5-4,2.5/3)[regular polygon, regular polygon sides=3,draw=black,fill=green,scale=0.35]; at (4.5+17.5,2.5/3)[regular polygon, regular polygon sides=3,draw=black,fill=green,scale=0.35];\nat (3.5+5.5,2.8); at (5.7,1.1); at (4+5.7,1.1); at (0.3-0.7,0.65); at (3.3-0.7,0.65); at (1+5.1,0.35); at (3+5.1,0.35); at (1+10.1,0.65+2.5/3); at (6+10.1,0.65+2.5/3); at (1.5+10.1,0.65); at (4.5+10.1,0.65); at (5.6+12,1.1); at (4+5.6+12,1.1); at (1.3,-0.5)F− 1; at (7.3,-0.5)F0; at (13.3,-0.5)F1; at (19.3,-0.5)F2;\n[ecoutecabb]\nSuppose, by contradiction, that there exists a 28-coloring *c* of F3. We begin by showing that *u* can not have the same color than the one of any vertex from *X*1, *X*2, *X*3 and *X*4. Afterward, we show that we have a contradiction in the case *u*, *v*1 and *v*2 have the same color. Consequently, *c*(*u*) ≠ *c*(*v*1) or *c*(*u*) ≠ *c*(*v*2), and *u* can not have the same color than the one of any vertex from either *B* or *B*ʹ (which both contain a vertex of every color).\nFirst, by Lemma [balancetonlemme].(iv), one vertex from {(1, − 1, − 2), (1.5, − 0.5, − 3), (0.5, − 0.5, − 3)} has the same color than *u*. Note that every vertex of {(1, − 1, − 2), (1.5, − 0.5, − 3), (0.5, − 0.5, − 3)} is at distance at most 3 from ( − 0.5, − 0.5, − 1), (2.5, − 0.5, − 1), (0, − 1, 0) and (2, − 1, 0). Consequently, *c*(*u*) ≠ *c*(*v*), if *v* ∈ *X*1.\nSecond, by Lemma [balancetonlemme].(iii), one vertex from {(1, − 1, 2), (1.5, − 0.5, 3), (0.5, − 0.5, 3)} has the same color than *u*. Note that every vertex of {(1, − 1, 2), (1.5, − 0.5, 3), (0.5, − 0.5, 3)} is at distance at most 3 from ( − 0.5, − 0.5, 1) and (2.5, − 0.5, 1). Consequently, *c*(*u*) ≠ *c*(*v*), if *v* ∈ *X*2.\nThird, by Lemma [balancetonlemme].(ii), one vertex from {( − 2, 2, 2), ( − 1.5, 2.5, 3), ( − 1.5, 1.5, 3)} has the same color than *u*. Note that every vertex of {( − 2, 2, 2), ( − 1.5, 2.5, 3), ( − 1.5, 1.5, 3)} is at distance at most 3 from ( − 1.5, 0.5, 1) and ( − 1, 0, 2). Consequently, *c*(*u*) ≠ *c*(*v*), if *v* ∈ *X*3.\nFourth, by Lemma [balancetonlemme].(i), one vertex from {(4, 2, 2), (3.5, 2.5, 3), (3.5, 1.5, 3)} has the same color than *u*. Note that every vertex of {(4, 2, 2), (3.5, 2.5, 3), (3.5, 1.5, 3)} is at distance at most 3 from (3.5, 0.5, 1) and (3, 0, 2). Consequently, *c*(*u*) ≠ *c*(*v*), if *v* ∈ *X*4.\nNow, by the previous arguments, we know that the color of *u* is different from the colors of the vertices of *B* \\ {*v*1} and of *B*ʹ \\ {*v*2}. Thus, since both *B* and *B*ʹ should contain a vertex of every color, we have *c*(*u*) = *c*(*v*1) and *c*(*u*) = *c*(*v*2). By Lemma [balancetonlemme].(i), one vertex from {(2, 0, 2), (1.5, 0.5, 3), (1.5, − 0.5, 3)} has the same color than *v*1. Note that *d*F(*v*2, (2, 0, 2)) = 2, *d*F(*v*2, (1.5, − 0.5, 3)) = 3 and *d*F(*v*2, (1.5, 0.5, 3)) = 3. Therefore, we have a contradiction with the fact that *c*(*v*1) = *c*(*v*2).\nTo conclude, *u* can not have the same color that any vertex of either *B* or *B*ʹ and since ∣*B*∣ = 28 and ∣*B*ʹ∣ = 28, F3 is not 28-colorable.\nChromatic number of the *d*th power of F0, 1 and F0, 2\n======================================================\nIn this section, we give bounds on the chromatic number of two subgraphs of F. These subgraphs are obtained by restricting the graph to the vertices of two or three consecutive layers of F. Note that we achieved to obtain the exact value of the chromatic number of the *d*th power of these subgraphs in more cases that for F.\nChromatic number of the *d*th power of F0, 1\n--------------------------------------------\nWe begin this subsection by calculating the size of a largest complete subgraph in F0, 1*d*. From this value, we derive the following lower bound on the chromatic number of F0, 1*d*.\nFor every positive integer *d*, we have *χ*(F0, 1*d*) ≥ (*d* + 1)2.\nFirst, suppose *d* is even. Let *A*ℓ = {(*i*, *j*, *k*) ∈ *V*(F0, 1)∣ *d*F((0, 0, 0)(*i*, *j*, *k*)) = ℓ}. We claim that ∣*A*ℓ∣ = 8ℓ, for ℓ ≥ 1. We proceed by induction to prove it. Note that ∣*A*1∣ = 8. Suppose, by induction, that ∣*A*ℓ∣ = 8ℓ. It can be easily remarked that both ∣*V*(F0) ∩ *A*ℓ + 1∣ = ∣*V*(F0) ∩ *A*ℓ∣ + 4 and ∣*V*(F1) ∩ *A*ℓ + 1∣ = ∣*V*(F1) ∩ *A*ℓ∣ + 4. Thus ∣*A*ℓ + 1∣ = 8ℓ + 8. By calculation ∣ ∪ *i* = 0ℓ*A**i*∣ = ∑*i* = 1ℓ(8*i*) + 1 = 4ℓ(ℓ + 1) + 1. Consequently, since diam(F0, 1[ ∪ *i* = 0ℓ*A**i*]) ≤ 2ℓ, we have: \n$$\\chi(\\mathcal{F}\\_{0,1}^{d})\\ge 4 \\frac{d}{2} \\left(\\frac{d}{2}+1\\right)+1=d(d+2)+1=(d+1)^2.$$ \nSecond, suppose *d* is odd. Let *D*0 = {(0, 0, 0), (1, 0, 0), (0.5, 0.5, 1), (0.5, − 0.5, 1)} and let *D*ℓ = {*u* ∈ *V*(F0, 1)∣ min*v* ∈ *D*0(*d*F(*u*, *v*)) = ℓ}, for ℓ ≥ 0. We claim that ∣*D*ℓ∣ = 8ℓ + 4, for ℓ ≥ 0. We proceed by induction to prove it. Note that ∣*D*0∣ = 4 and ∣*D*1∣ = 12. Suppose, by induction, that ∣*D*ℓ∣ = 8ℓ + 4. It can be easily remarked that both ∣*V*(F0) ∩ *D*ℓ + 1∣ = ∣*V*(F0) ∩ *D*ℓ∣ + 4 and ∣*V*(F1) ∩ *D*ℓ + 1∣ = ∣*V*(F1) ∩ *D*ℓ∣ + 4. Thus ∣*D*ℓ + 1∣ = 8ℓ + 8. By calculation, ∣ ∪ *i* = 0ℓ*D**i*∣ = ∑*i* = 0ℓ(8*i* + 4) = 4(ℓ + 1)2. Consequently, since diam(F0, 1[ ∪ *i* = 0ℓ*D**i*]) ≤ 2ℓ + 1, we have : \n$$\\chi(\\mathcal{F}\\_{0,1}^{d})\\ge 4\\left(\\frac{(d-1)}{2}+1\\right)^2=(d+1)^2.$$ \nIn the following proposition, we show that, in contrast with F*d*, the size of the largest complete subgraph in F0, 1*d* corresponds to the chromatic number of F0, 1*d*, for every integer *d*.\nFor every positive integer *d*, we have *χ*(F0, 1*d*) ≤ (*d* + 1)2.\nIt can be remarked that the coloring from Theorem [upbound] restricted to F0, 1*d* is a coloring using (*d* + 1)2 colors, when *d* is odd. Then, we can suppose that *d* is even and, consequently, that *d*/2 is an integer. We set the following function: ∀(*i*, *j*, *k*) ∈ *V*(F), \n$$c(i,j,k)) = i+(2d+1) j \\pmod{(d+1)^2}.$$ \nWe claim that *c* is a coloring function of F0, 1*d*.\nIt remains to prove that every two vertices (*i*, *j*, *k*) and (*i*ʹ, *j*ʹ, *k*ʹ) satisfying *c*((*i*, *j*, *k*)) = *c*((*i*ʹ, *j*ʹ, *k*ʹ)) are such that *d*F((*i*, *j*, *k*), (*i*ʹ, *j*ʹ, *k*ʹ)) > *d*. Without loss of generality, we suppose that *i*ʹ = 0, *j*ʹ = 0 and *k*ʹ = 0.\nNote that the vertices (*i*, *j*, *k*) at distance at most *d* from (0, 0, 0) are such that 1 ≤ ∣*i*∣ + ∣*j*∣ ≤ *d*. Also, note that *c*((0, 0, 0)) = *c*((*i*, *j*, *k*)) implies $i+(2d+1) j\\equiv 0\\pmod{(d+1)^2}$. Thus, it remains to show that for every possible value of *i* and *j*, 1 ≤ ∣*i*∣ + ∣*j*∣ ≤ *d*, we have $i+(2d+1) j\\not \\equiv 0\\pmod{(d+1)^2}$.\nIf ∣*j*∣ ≤ *d*/2, then − *d*2 − *d*/2 ≤ (2*d* + 1)∣*j*∣ ≤ *d*2 + *d*/2. Thus, since ∣*i*∣ + ∣*j*∣ ≤ *d*, we have − (*d* + 1)2 < − *d*2 − *d* ≤ *i* + (2*d* + 1)*j* ≤ *d*2 + *d* < (*d* + 1)2 and since ∣*i*∣ + ∣*j*∣ ≠ 0, we have $i+(2d+1) j\\not \\equiv 0\\pmod{(d+1)^2}$.\nMoreover, if *d*/2 + 1 ≤ ∣*j*∣ ≤ *d*, then *d*2 + 5*d*/2 + 1 ≤ (2*d* + 1)∣*j*∣ ≤ 2*d*2 + *d* and consequently, since ∣*i*∣ + ∣*j*∣ ≤ *d*, we have *d*2 + 2*d* + 2 ≤ *i* + (2*d* + 1)∣*j*∣ ≤ 2*d*2 + 3*d*/2 − 1. Since (*d* + 1)2 < *d*2 + 2*d* + 2 and 2*d*2 + 3*d*/2 − 1 < 2(*d* + 1)2, we have $i+(2d+1) j\\not \\equiv 0\\pmod{(d+1)^2}$.\nChromatic number of the *d*th power of F0, 2\n--------------------------------------------\nWe begin this subsection by giving a general upper bound (using Theorem [upbound]) on the chromatic number of F0, 2*d*.\nFor every positive integer *d*, we have *χ*(F0, 2*d*) ≤ 3⌈(*d* + 1)/2⌉2.\nIt can be remarked that the coloring from Theorem [upbound] restricted to F0, 2*d* is a coloring of F using 3⌈(*d* + 1)2/2⌉ colors.\nWe continue this subsection by determining the size of a largest complete subgraph in F0, 2*d*. From this value, we derive the following lower bound on the chromatic number of F0, 2*d*.\n[whitefire] For every positive integer *d*, we have *χ*(F0, 2*d*) ≥ 3(⌊(*d* + 1)2/2⌋) − 2.\nLet *A*ʹℓ = {(*i*, *j*, *k*) ∈ *V*(F0, 2)∣ *d*F0, 2((0, 0, 1), (*i*, *j*, *k*) = ℓ}. We prove by induction that ∣*A*ʹℓ∣ = 12ℓ, for ℓ ≥ 1. Note that we have ∣*A*ʹ0∣ = 1 and ∣*A*ʹ1∣ = 12. Suppose that ∣*A*ʹℓ∣ = 12ℓ. We can easily notice that, for each of the three layers of F0, 2, there are four more vertices in *A*ʹℓ + 1, than in *A*ʹℓ. Consequently, ∣*A*ʹℓ + 1∣ = 12ℓ + 12 = 12(ℓ + 1). Let *A*ℓ = ∪ 0 ≤ *i* ≤ ℓ*A*ʹ*i*. By calculation, ∣*A*ℓ∣ = ∑*i* = 0ℓ∣*A*ʹ*i*∣ = 1 + 12∑*i* = 1ℓ*i* = 1 + 6ℓ(ℓ + 1) = 6ℓ2 + 6ℓ + 1. Since diam(F0, 2[*A*ℓ]) = 2ℓ, we have *χ*(F0, 2*d*) ≥ 3/2*d*2 + 3*d* + 1 = 3⌈(*d* + 1)2/2⌉ − 1, for an even *d*.\nLet *B*ʹℓ = {(*i*, *j*, *k*) ∈ *V*(F0, 2)∣ *d*F(*u*, (*i*, *j*, *k*) = ℓ, *u* ∈ {(0, 0, 0), (1, 0, 0), (0.5, − 0.5, 1), (0.5, 0.5, 1)}}. We prove by induction that *B*ʹℓ = 12ℓ + 6, for ℓ ≥ 1. Note that we have ∣*B*ʹ0∣ = 4 and ∣*B*ʹ1∣ = 18. Suppose that ∣*B*ʹℓ∣ = 12ℓ + 6. We can easily notice that, for each of the three layers of F0, 2, there are four more vertices in ∣*B*ʹℓ + 1∣, than in ∣*B*ʹℓ∣. Consequently, ∣*B*ʹℓ + 1∣ = 12ℓ + 12 + 6 = 12(ℓ + 1) + 6. Let *B*ℓ = ∪ 0 ≤ *i* ≤ ℓ*B*ʹ*i*. By calculation, ∣*B*ℓ∣ = ∑*i* = 0ℓ∣*B*ʹ*i*∣ = 4 + 12∑*i* = 1ℓ*i* + 6ℓ = 4 + 6ℓ(ℓ + 1) + 6ℓ = 6ℓ2 + 12ℓ + 4. Since diam(F0, 2[*B*ℓ]) = 2ℓ + 1, we have *χ*(F0, 2*d*) ≥ 6/4(*d* − 1)2 + 6(*d* − 1) + 4 = 3/2*d*2 + 3*d* − 1/2 = 3⌈(*d* + 1)2/2⌉ − 2, for an odd *d*.\nNow, we prove that the lower bound from Proposition [whitefire] can be improved. In order to do this, we begin by proving three Lemmas that will be used in the proof of Theorem [sharknado].\n[le2661] If there exists a 23-coloring *c* of F0, 23, then for every integers *i* and *j*, *c*((*i*, *j*, 0)) = *c*((*i* + 3, *j*, 2)) or *c*((*i* + 3, *j*, 0)) = *c*((*i*, *j*, 2)).\n[scale=1.2]\n(0-1,2.5/3-0.5) – (2,2.5/3-0.5); (0-0.5,5/3-0.5) – (3-0.5,5/3-0.5); (0.5-0.5,2) – (3,2); (-1,2.5/3-0.5) – (0,2); (0,2.5/3-0.5) – (1,2); (1,2.5/3-0.5) – (2,2); (2,2.5/3-0.5) – (3,2);\n(0+3.5,0) – (2+3.5,0); (0.5+3.5,2.5/3) – (2.5+3.5,2.5/3); (1+3.5,5/3) – (3+3.5,5/3); (1.5+3.5,2.5) – (3.5+3.5,2.5); (0+3.5,0) – (1.5+3.5,2.5); (1+3.5,0) – (2.5+3.5,2.5); (2+3.5,0) – (3.5+3.5,2.5);\n(0-1+8.5,2.5/3-0.5) – (2+8.5,2.5/3-0.5); (0-0.5+8.5,5/3-0.5) – (3-0.5+8.5,5/3-0.5); (0.5-0.5+8.5,2) – (3+8.5,2); (-1+8.5,2.5/3-0.5) – (0+8.5,2); (0+8.5,2.5/3-0.5) – (1+8.5,2); (1+8.5,2.5/3-0.5) – (2+8.5,2); (2+8.5,2.5/3-0.5) – (3+8.5,2);\n(0,2.5/3-0.5) – (1,2); (1,2.5/3-0.5) – (2,2); (1-1,2.5/3-0.5) – (1,2.5/3-0.5); (-0.5,5/3-0.5) – (3-0.5,5/3-0.5); (1.5-0.5,2) – (2,2);\n(0.5+3.5,2.5/3) – (1+3.5,5/3); (1+3.5,0) – (2.5+3.5,2.5); (2.5+3.5,2.5/3) – (3+3.5,5/3); (1.5/3+3.5,2.5/3) – (2+1.5/3+3.5,2.5/3); (1+3.5,5/3) – (3+3.5,5/3);\n(0+8.5,2.5/3-0.5) – (1+8.5,2); (1+8.5,2.5/3-0.5) – (2+8.5,2); (1-1+8.5,2.5/3-0.5) – (1+8.5,2.5/3-0.5); (1-0.5+8.5,5/3-0.5) – (2-0.5+8.5,5/3-0.5); (1.5-0.5+8.5,2) – (2+8.5,2);\nat (0,2.5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (1,2.5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (-0.5,5/3-0.5)[regular polygon, regular polygon sides=4,draw=black,fill=red,scale=0.5]; at (0.5,5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (1.5,5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (2.5,5/3-0.5) [regular polygon, regular polygon sides=4,draw=black,fill=red,scale=0.5]; at (1,2)[circle,draw=black,fill=black,scale=0.5]; at (2,2)[circle,draw=black,fill=black,scale=0.5];\nat (2.5+3.5,2.5)[circle,draw=black,fill=black,scale=0.5]; at (3+3.5,5/3)[circle,draw=black,fill=black,scale=0.5]; at (2.5+3.5,2.5/3)[circle,draw=black,fill=black,scale=0.5]; at (3+3.5-1,5/3)[circle,draw=black,fill=black,scale=0.5]; at (2.5+3.5-1,2.5/3)[circle,draw=black,fill=black,scale=0.5]; at (3+3.5-2,5/3)[circle,draw=black,fill=black,scale=0.5]; at (4.5+3.5-4,2.5/3)[circle,draw=black,fill=black,scale=0.5]; at (1+3.5,0)[circle,draw=black,fill=black,scale=0.5]; at (0+8.5,2.5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (1+8.5,2.5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (-0.5+8.5,5/3-0.5)[regular polygon, regular polygon sides=4,draw=black,fill=red,scale=0.5]; at (0.5+8.5,5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (1.5+8.5,5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (2.5+8.5,5/3-0.5)[regular polygon, regular polygon sides=4,draw=black,fill=red,scale=0.5]; at (1+8.5,2)[circle,draw=black,fill=black,scale=0.5]; at (2+8.5,2)[circle,draw=black,fill=black,scale=0.5];\nat (0.95-1.7,0.6+2.5/3); at (3.95-1.7,0.6+2.5/3);\nat (0.95+8.5-1.7,0.6+2.5/3); at (3.95+8.5-1.7,0.6+2.5/3);\nat (0.5,-0.5)F0; at (4.5,-0.5)F1; at (9,-0.5)F2;\n[blackninja]\nLet *c* be a 23-coloring of F0, 2 and let *u*1 = (*i*, *j*, 0), *u*2 = (*i*, *j*, 2), *v*1 = (*i* + 3, *j*, 0) and *v*2 = (*i* + 3, *j*, 2). Suppose that *c*(*u*1) ≠ *c*(*v*2). Let *B* = *T*((*i* + 1, *j*, 0), (*i* + 2, *j*, 0), (*i* + 1.5, *j* − 0.5, 1), (*i* + 1.5, *j* + 0.5, 1)). Figure [blackninja] illustrates the set *B* and the vertices *u*1, *u*2, *v*1 and *v*2. Also, note that *B* induces a subgraph of diameter 3 in F0, 2 and that ∣*B*∣ = 22. For this reason, 22 colors are required to color *B*. Since *v*2 is at distance at most 3 of every vertex of *B* \\ {*u*1}, 23 colors are required to color *B* ∪ {*v*2}. Also, note that *u*2 is at distance at most 3 of every vertex of *B* \\ {*v*1} and that *d*F0, 2(*u*2, *v*2) = 3. Since *u*2 is at distance at most 3 of every vertex of *B* \\ {*v*1} ∪ {*v*2}, the only possibility is that *c*(*u*2) = *c*(*v*1) which conclude our proof. The proof is totally similar in the case *c*(*v*1) ≠ *c*(*u*2).\nThe following Lemma is an intermediate step in order to prove Lemma [le2667] that will be used several times in the proof of Theorem [sharknado].\n[le2665] If there exists a 23-coloring *c* of F0, 23 and integers *i* and *j* such that *c*((*i*, *j*, 0)) = *c*((*i* + 3, *j*, 2)) and *c*((*i* + 3, *j*, 0)) = *c*((*i*, *j*, 2)), then *c*((*i*, *j* + 1, 0)) = *c*((*i* + 3, *j* + 1, 2)) and *c*((*i* + 3, *j* + 1, 0)) = *c*((*i*, *j* + 1, 2)).\nLet *u*1 = (*i*, *j*, 0), *u*2 = (*i*, *j*, 2), *v*1 = (*i* + 3, *j*, 0), *v*2 = (*i* + 3, *j*, 2), *u*ʹ1 = (*i*, *j* + 1, 0), *u*ʹ2 = (*i*, *j* + 1, 2), *v*ʹ1 = (*i* + 3, *j* + 1, 0) and *v*ʹ2 = (*i* + 3, *j* + 1, 2). Let *c* be a 23-coloring of F0, 23 such that *c*(*u*1) = *c*(*v*2) and *c*(*u*2) = *c*(*v*1). Suppose by contradiction that *c*(*u*ʹ1) ≠ *c*(*v*ʹ2) or *c*(*v*ʹ1) ≠ *c*(*u*ʹ2). Let *B* = *T*((*i* + 1, *j* + 1, 0), (*i* + 2, *j* + 1, 0), (*i* + 1.5, *j* + 0.5, 1), (*i* + 1.5, *j* + 1.5, 1)). Note that *B* induces in F0, 2 a subgraph of diameter 3 containing 22 vertices. The following is true:\n* *v*ʹ2 is at distance at most 3 from any vertex of *B* \\ {*u*ʹ1};\n* *u*ʹ2 is at distance at most 3 from any vertex of *B* \\ {*v*ʹ1};\n* every vertex of (*B* \\ {(*i* + 1.5, *j* + 2.5, 1)}) ∪ {*u*ʹ2, *v*ʹ2} is at distance at most 3 from either *u*1 or *v*2 and at distance at most 3 from either *v*1 or *u*2.\nFigure [transmorpher] illustrates why iii) holds. Note that *B* ∪ {*u*ʹ2, *v*ʹ2} contains, by i) and ii), vertices of 23 different colors. Let *a* = *c*(*u*1) = *c*(*v*2) and *b* = *c*(*v*1) = *c*(*u*2).\n[scale=1.2]\n(0-1,2.5/3-0.5) – (2,2.5/3-0.5); (0-0.5,5/3-0.5) – (3-0.5,5/3-0.5); (0.5-0.5,2) – (3,2); (-1,2.5/3-0.5) – (0,2); (0,2.5/3-0.5) – (1,2); (1,2.5/3-0.5) – (2,2); (2,2.5/3-0.5) – (3,2);\n(0+3.5,0) – (2+3.5,0); (0.5+3.5,2.5/3) – (2.5+3.5,2.5/3); (1+3.5,5/3) – (3+3.5,5/3); (1.5+3.5,2.5) – (3.5+3.5,2.5); (0+3.5,0) – (1.5+3.5,2.5); (1+3.5,0) – (2.5+3.5,2.5); (2+3.5,0) – (3.5+3.5,2.5);\n(0-1+8.5,2.5/3-0.5) – (2+8.5,2.5/3-0.5); (0-0.5+8.5,5/3-0.5) – (3-0.5+8.5,5/3-0.5); (0.5-0.5+8.5,2) – (3+8.5,2); (-1+8.5,2.5/3-0.5) – (0+8.5,2); (0+8.5,2.5/3-0.5) – (1+8.5,2); (1+8.5,2.5/3-0.5) – (2+8.5,2); (2+8.5,2.5/3-0.5) – (3+8.5,2);\n(0,2.5/3-0.5) – (1,2); (1,2.5/3-0.5) – (2,2); (1-1,2.5/3-0.5) – (1,2.5/3-0.5); (-0.5,5/3-0.5) – (3-0.5,5/3-0.5); (1.5-0.5,2) – (2,2);\n(0.5+3.5,2.5/3) – (1+3.5,5/3); (1+3.5,0) – (2.5+3.5,2.5); (2.5+3.5,2.5/3) – (3+3.5,5/3); (1.5/3+3.5,2.5/3) – (2+1.5/3+3.5,2.5/3); (1+3.5,5/3) – (3+3.5,5/3);\n(0+8.5,2.5/3-0.5) – (1+8.5,2); (1+8.5,2.5/3-0.5) – (2+8.5,2); (1-1+8.5,2.5/3-0.5) – (1+8.5,2.5/3-0.5); (-0.5+8.5,5/3-0.5) – (3-0.5+8.5,5/3-0.5); (1.5-0.5+8.5,2) – (2+8.5,2);\nat (0-1,2.5/3-0.5)[regular polygon, regular polygon sides=4,draw=black,fill=red,scale=0.5]; at (3-1,2.5/3-0.5)[regular polygon, regular polygon sides=4,draw=black,fill=red,scale=0.5]; at (0,2.5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (1,2.5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (-0.5,5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (0.5,5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (1.5,5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (2.5,5/3-0.5) [regular polygon, regular polygon sides=3,draw=black,fill=green,scale=0.35]; at (1,2)[circle,draw=black,fill=black,scale=0.5]; at (2,2)[regular polygon, regular polygon sides=3,draw=black,fill=green,scale=0.35];\nat (3+3.5,5/3)[regular polygon, regular polygon sides=3,draw=black,fill=green,scale=0.35]; at (2.5+3.5,2.5/3)[regular polygon, regular polygon sides=3,draw=black,fill=green,scale=0.35]; at (3+3.5-1,5/3)[circle,draw=black,fill=black,scale=0.5]; at (2.5+3.5-1,2.5/3)[circle,draw=black,fill=black,scale=0.5]; at (3+3.5-2,5/3)[circle,draw=black,fill=black,scale=0.5]; at (4.5+3.5-4,2.5/3)[circle,draw=black,fill=black,scale=0.5]; at (1+3.5,0)[circle,draw=black,fill=black,scale=0.5];\nat (8.5-1,2.5/3-0.5)[regular polygon, regular polygon sides=4,draw=black,fill=red,scale=0.5]; at (3+7.5,2.5/3-0.5)[regular polygon, regular polygon sides=4,draw=black,fill=red,scale=0.5]; at (0+8.5,2.5/3-0.5)[regular polygon, regular polygon sides=3,draw=black,fill=green,scale=0.35]; at (1+8.5,2.5/3-0.5)[regular polygon, regular polygon sides=3,draw=black,fill=green,scale=0.35]; at (-0.5+8.5,5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (0.5+8.5,5/3-0.5)[regular polygon, regular polygon sides=3,draw=black,fill=green,scale=0.35]; at (1.5+8.5,5/3-0.5)[regular polygon, regular polygon sides=3,draw=black,fill=green,scale=0.35]; at (2.5+8.5,5/3-0.5)[regular polygon, regular polygon sides=3,draw=black,fill=green,scale=0.35]; at (1+8.5,2)[circle,draw=black,fill=black,scale=0.5]; at (2+8.5,2)[regular polygon, regular polygon sides=3,draw=black,fill=green,scale=0.35];\nat (0.95-1.7,0.6+2.5/3); at (3.95-1.7,0.6+2.5/3);\nat (0.95+8.5-1.7,0.6+2.5/3); at (3.95+8.5-1.7,0.6+2.5/3); at (6,2.7);\nat (0.95-2.2,0.6); at (3.95-2.2,0.6);\nat (0.95+8-1.7,0.6); at (3.95+8-1.7,0.6);\nat (0.5,-0.5)F0; at (4.5,-0.5)F1; at (9,-0.5)F2;\n[transmorpher]\nBy iii), both the colors *a* and *b* are not used for vertices of (*B* \\ {(*i* + 1.5, *j* + 2.5, 1)}) ∪ {*u*ʹ2, *v*ʹ2}. Moreover, since *c*((*i* + 1.5, *j* + 2.5, 1)) ≠ *a* or *c*((*i* + 1.5, *j* + 2.5, 1)) ≠ *b* and by iii), the color *a* or *b* is different than each of the 23 other colors and we have a contradiction with the fact that F0, 23 is 23-colorable.\nNote that the following Lemma is really useful since it implies that if *B* = *T*((*i*, *j*, 0), (*i* + 1, *j*, 0), (*i* + 0.5, *j* − 0.5, 1), (*i* + 0.5, *j* + 0.5, 1)) and there exists a 23-coloring *c* of F0, 23, then *B* ∪ {(*i* − 1, *j*, 2), (*i* + 2, *j*, 2)} contains 23 different colors. This is due to the fact that (*i* − 1, *j*, 2) is at distance at most 3 of every vertex of *B* \\ {(*i* + 2, *j*, 0)} and that (*i* + 2, *j*, 2) is at distance at most 3 of every vertex of *B* \\ {(*i* − 1, *j*, 0)}.\n[le2667] There exists no 23-coloring *c* of F0, 23 for which there exists integers *i* and *j* such that *c*((*i*, *j*, 0)) = *c*((*i* + 3, *j*, 2)) and *c*((*i* + 3, *j*, 0)) = *c*((*i*, *j*, 2)).\nLet *u*1 = (*i*, *j*, 0), *u*2 = (*i*, *j*, 2), *v*1 = (*i* + 3, *j*, 0) and *v*2 = (*i* + 3, *j*, 2). Suppose, by contradiction, that there is a 23-coloring *c* of F0, 23 such that *c*(*u*1) = *c*(*v*2) and *c*(*v*1) = *c*(*u*2). By Lemma [le2665], we have *c*((*i*, *j* + 1, 0)) = *c*((*i* + 3, *j* + 1, 2)) and *c*((*i* + 3, *j* + 1, 0)) = *c*((*i*, *j* + 1, 2)) and also *c*((*i*, *j* + 2, 0)) = *c*((*i* + 3, *j* + 2, 2)) and *c*((*i* + 3, *j* + 2, 0)) = *c*((*i*, *j* + 2, 2)). Let *B* = *T*((*i* + 1, *j* + 1, 0), (*i* + 2, *j* + 1, 0), (*i* + 1.5, *j* + 0.5, 1), (*i* + 1.5, *j* + 1.5, 1)).\nNote that *B* induces in F0, 2 a subgraph of diameter 3 containing 22 vertices. The following is true :\n* every vertex of *B* \\ {(*i* + 1.5, *j* − 2.5, 1)} is at distance at most 3 from either *u*1 or *v*2 and at most 3 from either *v*1 or *u*2;\n* every vertex of *B* \\ {(*i* + 1.5, *j* + 0.5, 1)} is at distance at most 3 from either (*i*, *j* + 2, 0) or (*i* + 3, *j* + 2, 2) and at most 3 from either (*i* + 3, *j* + 2, 0) or (*i*, *j* + 2, 2).\nLet *a* = *c*(*u*1), *b* = *c*(*u*2), *a*ʹ = *c*((*i*, *j* + 2, 0)) and *b*ʹ = ((*i*, *j* + 2, 2)). Since *u*1, *u*2, (*i*, *j* + 2, 0) and (*i*, *j*, 2) are pairwise at distance at most 3, the colors *a*, *b*, *a*ʹ, *b*ʹ are all different. By i), either *a* or *b* is a color not used for vertices in *B*. Also, by ii), either *a*ʹ or *b*ʹ is a color not used for vertices in *B*. Therefore, since *B* contains vertices of 22 colors, we have to use two other colors. Consequently, we have a contradiction with the fact that F0, 23 is 23-colorable.\nWe finish this paper by determining the exact value of the chromatic number of F0, 23. Note that the exact value of the chromatic number of F3 remains undetermined.\n[sharknado] We have *χ*(F0, 23) ≥ 24.\nSuppose by contradiction that there exists a 23-coloring *c* of F0, 23. Let *u*1 = (0, 0, 0), *u*2 = (0, 0, 2), *v*1 = (3, 0, 0), *v*2 = (3, 0, 2), *x*1 = (0, 3, 0), *x*2 = (0, 3, 2), *y*1 = ( − 3, 0, 0) and *y*2 = ( − 3, 0, 2). Let *B* = *T*((1, 1, 0), (2, 1, 0), (1.5, 0.5, 1), (1.5, 1.5, 1)) and let *B*ʹ = *T*(( − 2, 1, 0), ( − 1, 1, 0), ( − 1.5, 0.5, 1), ( − 1.5, 1.5, 1)).\nWe first show that the following two cases can not occur:\n[scale=0.75]\n(0-1,2.5/3-0.5) – (5,2.5/3-0.5); (0-0.5,5/3-0.5) – (6-0.5,5/3-0.5); (0.5-0.5,2) – (6,2); (0.5,2+2.5/3) – (6.5,2+2.5/3); (-1,2.5/3-0.5) – (0.5,2+2.5/3); (0,2.5/3-0.5) – (1.5,2+2.5/3); (1,2.5/3-0.5) – (2.5,2+2.5/3); (2,2.5/3-0.5) – (3.5,2+2.5/3); (3,2.5/3-0.5) – (4.5,2+2.5/3); (4,2.5/3-0.5) – (5.5,2+2.5/3); (5,2.5/3-0.5) – (6.5,2+2.5/3);\n(0+6.5,0) – (5+6.5,0); (0.5+6.5,2.5/3) – (5.5+6.5,2.5/3); (1+6.5,5/3) – (6+6.5,5/3); (1.5+6.5,2.5) – (6.5+6.5,2.5); (0+6.5,0) – (1.5+6.5,2.5); (1+6.5,0) – (2.5+6.5,2.5); (2+6.5,0) – (3.5+6.5,2.5); (3+6.5,0) – (4.5+6.5,2.5); (4+6.5,0) – (5.5+6.5,2.5); (5+6.5,0) – (6.5+6.5,2.5);\n(0-1+14.5,2.5/3-0.5) – (5+14.5,2.5/3-0.5); (0-0.5+14.5,5/3-0.5) – (6-0.5+14.5,5/3-0.5); (0.5-0.5+14.5,2) – (6+14.5,2); (0.5+14.5,2+2.5/3) – (6.5+14.5,2+2.5/3); (-1+14.5,2.5/3-0.5) – (0.5+14.5,2+2.5/3); (0+14.5,2.5/3-0.5) – (1.5+14.5,2+2.5/3); (1+14.5,2.5/3-0.5) – (2.5+14.5,2+2.5/3); (2+14.5,2.5/3-0.5) – (3.5+14.5,2+2.5/3); (3+14.5,2.5/3-0.5) – (4.5+14.5,2+2.5/3); (4+14.5,2.5/3-0.5) – (5.5+14.5,2+2.5/3); (5+14.5,2.5/3-0.5) – (6.5+14.5,2+2.5/3);\n(0+3,2.5/3-0.5) – (1+3,2); (1+3,2.5/3-0.5) – (2+3,2); (1-1+3,2.5/3-0.5) – (1+3,2.5/3-0.5); (-0.5+3,5/3-0.5) – (3-0.5+3,5/3-0.5); (1.5-0.5+3,2) – (2+3,2); (0.5+6.5+3,2.5/3) – (1+6.5+3,5/3); (1+6.5+3,0) – (2.5+6.5+3,2.5); (2.5+6.5+3,2.5/3) – (3+6.5+3,5/3); (1.5/3+6.5+3,2.5/3) – (2+1.5/3+6.5+3,2.5/3); (1+6.5+3,5/3) – (3+6.5+3,5/3); (0+14.5+3,2.5/3-0.5) – (1+14.5+3,2); (1+14.5+3,2.5/3-0.5) – (2+14.5+3,2); (1-1+14.5+3,2.5/3-0.5) – (1+14.5+3,2.5/3-0.5); (-0.5+14.5+3,5/3-0.5) – (3-0.5+14.5+3,5/3-0.5); (1.5-0.5+14.5+3,2) – (2+14.5+3,2);\n(0,2.5/3-0.5) – (1,2); (1,2.5/3-0.5) – (2,2); (1-1,2.5/3-0.5) – (1,2.5/3-0.5); (-0.5,5/3-0.5) – (3-0.5,5/3-0.5); (1.5-0.5,2) – (2,2); (0.5+6.5,2.5/3) – (1+6.5,5/3); (1+6.5,0) – (2.5+6.5,2.5); (2.5+6.5,2.5/3) – (3+6.5,5/3); (1.5/3+6.5,2.5/3) – (2+1.5/3+6.5,2.5/3); (1+6.5,5/3) – (3+6.5,5/3); (0+14.5,2.5/3-0.5) – (1+14.5,2); (1+14.5,2.5/3-0.5) – (2+14.5,2); (1-1+14.5,2.5/3-0.5) – (1+14.5,2.5/3-0.5); (-0.5+14.5,5/3-0.5) – (3-0.5+14.5,5/3-0.5); (1.5-0.5+14.5,2) – (2+14.5,2);\nat (0-1,2.5/3-0.5)[regular polygon, regular polygon sides=6,draw=black,fill=yellow,scale=0.5]; at (3-1,2.5/3-0.5)[regular polygon, regular polygon sides=6,draw=black,fill=yellow,scale=0.5]; at (6-1,2.5/3-0.5)[regular polygon, regular polygon sides=6,draw=black,fill=yellow,scale=0.5]; at (0,2.5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (1,2.5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (3,2.5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (4,2.5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (-0.5,5/3-0.5)[regular polygon, regular polygon sides=3,draw=black,fill=green,scale=0.35]; at (0.5,5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (1.5,5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (2.5,5/3-0.5) [circle,draw=black,fill=black,scale=0.5]; at (3.5,5/3-0.5)[circle,draw=black,fill=black,scale=0.5]; at (4.5,5/3-0.5) [circle,draw=black,fill=black,scale=0.5]; at (5.5,5/3-0.5)[regular polygon, regular polygon sides=4,draw=black,fill=red,scale=0.5]; at (1,2)[regular polygon, regular polygon sides=3,draw=black,fill=green,scale=0.35]; at (2,2)[circle,draw=black,fill=black,scale=0.5]; at (4,2)[circle,draw=black,fill=black,scale=0.5]; at (5,2)[regular polygon, regular polygon sides=4,draw=black,fill=red,scale=0.5];\nat (3+6.5,5/3)[circle,draw=black,fill=black,scale=0.5]; at (2.5+6.5,2.5/3)[circle,draw=black,fill=black,scale=0.5]; at (3+6.5-1,5/3)[circle,draw=black,fill=black,scale=0.5]; at (2.5+6.5-1,2.5/3)[circle,draw=black,fill=black,scale=0.5]; at (3+6.5-2,5/3)[regular polygon, regular polygon sides=3,draw=black,fill=green,scale=0.35]; at (4.5+6.5-4,2.5/3)[regular polygon, regular polygon sides=3,draw=black,fill=green,scale=0.35]; at (1+6.5,0)[circle,draw=black,fill=black,scale=0.5]; at (3+9.5,5/3)[regular polygon, regular polygon sides=4,draw=black,fill=red,scale=0.5]; at (2.5+9.5,2.5/3)[regular polygon, regular polygon sides=4,draw=black,fill=red,scale=0.5]; at (3+9.5-1,5/3)[circle,draw=black,fill=black,scale=0.5]; at (2.5+9.5-1,2.5/3)[circle,draw=black,fill=black,scale=0.5]; at (3+9.5-2,5/3)[circle,draw=black,fill=black,scale=0.5]; at (4.5+9.5-4,2.5/3)[circle,draw=black,fill=black,scale=0.5]; at (1+9.5,0)[circle,draw=black,fill=black,scale=0.5];\nat (0-1+14.5,2.5/3-0.5)[regular polygon, regular polygon sides=6,draw=black,fill=yellow,scale=0.5]; at (3-1+14.5,2.5/3-0.5)[regular polygon, regular polygon sides=6,draw=black,fill=yellow,scale=0.5]; at (6-1+14.5,2.5/3-0.5)[regular polygon, regular polygon sides=6,draw=black,fill=yellow,scale=0.5]; at (0+14.5,2.5/3-0.5)[regular polygon, regular polygon sides=3,draw=black,fill=green,scale=0.35]; at (1+14.5,2.5/3-0.5)[regular polygon, regular polygon sides=3,draw=black,fill=green,scale=0.35]; at (3+14.5,2.5/3-0.5)[regular polygon, regular polygon sides=4,draw=black,fill=red,scale=0.5]; at (4+14.5,2.5/3-0.5)[regular polygon, regular polygon sides=4,draw=black,fill=red,scale=0.5]; at (-0.5+14.5,5/3-0.5)[regular polygon, regular polygon sides=3,draw=black,fill=green,scale=0.35]; at (0.5+14.5,5/3-0.5)[regular polygon, regular polygon sides=3,draw=black,fill=green,scale=0.35]; at (1.5+14.5,5/3-0.5)[regular polygon, regular polygon sides=3,draw=black,fill=green,scale=0.35]; at (2.5+14.5,5/3-0.5) [circle,draw=black,fill=black,scale=0.5]; at (3.5+14.5,5/3-0.5)[regular polygon, regular polygon sides=4,draw=black,fill=red,scale=0.5]; at (4.5+14.5,5/3-0.5) [regular polygon, regular polygon sides=4,draw=black,fill=red,scale=0.5]; at (5.5+14.5,5/3-0.5)[regular polygon, regular polygon sides=4,draw=black,fill=red,scale=0.5]; at (1+14.5,2)[regular polygon, regular polygon sides=3,draw=black,fill=green,scale=0.35]; at (2+14.5,2)[circle,draw=black,fill=black,scale=0.5]; at (4+14.5,2)[circle,draw=black,fill=black,scale=0.5]; at (5+14.5,2)[regular polygon, regular polygon sides=4,draw=black,fill=red,scale=0.5]; at (3.5,2+2.5/3)[regular polygon, regular polygon sides=6,draw=black,fill=yellow,scale=0.5]; at (3.5+14.5,2+2.5/3)[regular polygon, regular polygon sides=6,draw=black,fill=yellow,scale=0.5];\nat (0.9-2.2,0.65); at (3.9-2.2,0.65); at (6.9-2.2,0.65); at (3.2,2+2.5/3+0.35);\nat (0.9-2.2+14.5,0.65); at (3.9-2.2+14.5,0.65); at (6.9-2.2+14.5,0.65); at (3.2+14.5,2+2.5/3+0.35);\nat (2.5+6.5,2.8); at (5.5+6.5,2.8);\nat (2,-0.5)F0; at (16.5,-0.5)F2; at (9,-0.5)F1;\n[twoheadedshark]\nCase 1: *c*(*u*1) = *c*(*v*2) = *c*(*x*2) or *c*(*u*2) = *c*(*v*1) = *c*(*x*1).\nBy Lemma [le2667], since *c*((0, 1, 0)) ≠ *c*((3, 1, 2)) or *c*((0, 1, 2)) ≠ *c*((3, 1, 0), one vertex among (0, 1, 2) and (3, 1, 2) has a color not given to the vertices of *B* (since (0, 1, 0) is at distance at most 3 of every vertex of *B* \\ {(3, 1, 2)} and that (3, 1, 0) is at distance at most 3 of every vertex of *B* \\ {(0, 1, 2)}). Suppose *c*(*u*1) = *c*(*v*2) = *c*(*x*2). It can be easily checked that every vertex of *B* ∪ {(0, 1, 2), (3, 1, 2)} is at distance at most 3 of (at least) one vertex among {*u*1, *v*2, *x*2}. Figure [twoheadedshark] illustrates the fact that every vertex of *B* \\ {(1.5, 2.5, 1)} ∪ {(0, 1, 2), (3, 1, 2)} is at distance at most 3 of one vertex among {*u*1, *v*2}. It can be easily verified that *d*F0, 2(*x*2, (1.5, 2.5, 1)) ≤ 3.\nThus, *c*(*u*1) is different from the 23 used colors. Therefore, we obtain a contradiction.\nIn the case *c*(*u*2) = *c*(*v*1) = *c*(*x*1) it can be remarked that every vertex of *B* ∪ {(0, 1, 2), (3, 1, 2)} is at distance at most 3 of (at least) one vertex among {*u*2, *v*1, *x*1} and, as previously, we obtain a contradiction since *B* ∪ {(0, 1, 2), (3, 1, 2)} contain vertices of 23 different colors.\nCase 2: *c*(*u*1) = *c*(*v*2) = *c*(*y*2) or *c*(*u*2) = *c*(*v*1) = *c*(*y*1).\nSuppose that *c*(*u*1) = *c*(*v*2) = *c*(*y*2). It can be easily verified that every vertex of (*B* \\ {(1.5, 2.5, 1)}) ∪ {(0, 1, 2), (3, 1, 2)} is at distance at most 3 of (at least) one vertex among {*u*1, *v*2} and that every vertex of (*B*ʹ \\ {( − 1.5, 2.5, 1)}) ∪ {(0, 1, 2), ( − 3, 1, 2)} is at distance at most 3 of one vertex among {*u*1, *y*1}. In Figure [twoheadedshark], we illustrates the fact that every vertex of (*B* \\ {(1.5, 2.5, 1)}) ∪ {(0, 1, 2), (3, 1, 2)} is at distance at most 3 of one vertex among {*u*1, *v*2}. By Lemma [le2667], one vertex among (0, 1, 2) and (3, 1, 2) has a color not given to the vertices of *B* and one vertex among (0, 1, 2) and ( − 3, 1, 2) has a color not given to the vertices of *B*ʹ. Since both *B* ∪ {(0, 1, 2), (3, 1, 2)} and *B*ʹ ∪ {(0, 1, 2), ( − 3, 1, 2)} contain vertices of 23 different colors, we have *c*(*u*1) = *c*((1.5, 2.5, 1)) and *c*(*u*1) = *c*(( − 1.5, 2.5, 1)). However, since *d*F0, 2((1.5, 2.5, 1), ( − 1.5, 2.5, 1)) = 3, we have a contradiction.\nIn the case *c*(*u*2) = *c*(*v*1) = *c*(*y*1), since every vertex of (*B* \\ {(1.5, 2.5, 1)}) ∪ {(0, 1, 2), (3, 1, 2)} is at distance at most 3 of one vertex among {*u*2, *v*1} and every vertex of (*B*ʹ \\ {( − 1.5, 2.5, 1)}) ∪ {(0, 1, 2), ( − 3, 1, 2)} is at distance at most 3 of one vertex among {*u*2, *y*1}, we have also a contradiction since, as previously, both *B* ∪ {(0, 1, 2), (3, 1, 2)} and *B*ʹ ∪ {(0, 1, 2), ( − 3, 1, 2)} contain vertices of 23 different colors (this implies *c*(*u*2) = *c*((1.5, 2.5, 1)) and *c*(*u*2) = *c*(( − 1.5, 2.5, 1)).\nBy Lemma [le2661], either *c*(*u*1) = *c*(*v*2) or *c*(*u*2) = *c*(*v*1) holds. First, suppose *c*(*u*1) = *c*(*v*2). By Case 1, we have *c*(*u*1) ≠ *c*(*x*2) and by Case 2, we have *c*(*u*1) ≠ *c*(*y*2). Consequently, by Lemma [le2661], we have *c*(*u*2) = *c*(*x*1) = *c*(*y*1) and by symmetry and Lemma [le2661], we can not have both *c*(*u*1) ≠ *c*(*y*2) and *c*(*u*2) ≠ *c*(*y*1) and a contradiction with Case 1. Second suppose *c*(*u*2) = *c*(*v*1). By Case 1, we have *c*(*u*2) ≠ *c*(*x*1) and by Case 2, we have *c*(*u*2) ≠ *c*(*y*1). Consequently, by Lemma [le2661], we have *c*(*u*1) = *c*(*x*2) = *c*(*y*2), and a contradiction with Case 1.\nFinally, we obtain that F0, 23 is not 23-colorable.\nWe have *χ*(F0, 23) = 24.\nAcknowledgments\n===============\nThis work was supported by the French “Investissements d’Avenir” program, project ISITE-BFC (contract ANR-15-IDEX-03).\n99 S. Bjornholm, Clusters, condensed matter in embryonic form, *Contemp. Phys.* **31**, 309–324, 1990. J. Bourgeois, S. Copen Goldstein, Distributed intelligent MEMS: progresses and perspectives, *IEEE Systems Journal* **9** (3), 1057–1068, 2015. J. J. Daymude, R. Gmyr, A. W. Richa, C. Scheideler, T. Strothmann, Improved leader election for self-organizing programmable matter, *ALGOSENSORS 2017: Algorithms for Sensor Systems*, 127–140, 2017. G. Fertin, E. Godard, A. Raspaud, Acyclic and *k*-distance coloring of the grid, *Inform. Process. Lett.* **87** (1)51–58, 2003. P. Jacko, S. Jendrol’, Distance coloring of the hexagonal lattice, *Discuss. Math. Graph Theory* **25**, 151–166, 2005. F. Kramer, H. Kramer, A survey on the distance-colouring of graphs, *Discrete Math.* **308**, 422–426, 2008. B. Piranda, J. Bourgeois, Geometrical Study of a Quasi-spherical Module for Building Programmable Matter, *Distributed Autonomous Robotic Systems*, 347–400, 2018. A. Sevcikova, Distant chromatic number of the planar graphs, *Manuscript*, Safarik University, 2001. P. Šparl, R. Witkowski, J. Žerovnik, Multicoloring of cannonball graphs, *Ars Mathematica Contemporanea* **10** (1), 31–44, 2016.\n"},"data_id":{"kind":"string","value":"arxiv_0185581"}}}],"truncated":false,"partial":true},"paginationData":{"pageIndex":463,"numItemsPerPage":100,"numTotalItems":46503,"offset":46300,"length":100}},"jwt":"eyJhbGciOiJFZERTQSIsImtpZCI6IjVHZDBvd0g5MTM2eDZjc1FvbE1zcktNWUZoRVFoUm5PVVVybEpjOUhaUEEifQ.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc4NjEzNzk5NiwianRpIjoiZDExZDNiZDAtM2NkMi00YjExLWFjNzctMzU2MTNmMDExNzBmIiwic3ViIjoiL2RhdGFzZXRzL0x5dW4wOTEyL0xvbmdBQkMiLCJleHAiOjE3ODYxNDE1OTYsImlzcyI6Imh0dHBzOi8vaHVnZ2luZ2ZhY2UuY28ifQ.ujso8t4WPlTShzpyio6fc0R92f2gufe0d1oXtqd-AXeRwNSj3-V8BF0ycnkMuuAbHISLs65C7USDpTDMCJtuDA","displayUrls":true,"splitSizeSummaries":[{"config":"default","split":"train","numRows":46503,"numBytesParquet":2327876865}]},"discussionsStats":{"closed":0,"open":2,"total":2},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
content stringlengths 34.5k 188k | data_id stringlengths 13 13 |
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The importance of galaxy interactions in triggering type II quasar activity
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[firstpage]
We present deep Gemini GMOS-S optical broad-band images for a complete sample of 20 SDSS selected type II quasars taken from, with redshifts in the range 0.... | arxiv_0185184 |
as the unseen classes, for each of which 5 labeled reviews are given, and NELL is used as the auxiliary KG.
### Question Answering
Zero-shot and few-shot question answering (QA)[11](#fn11) started to attract wide attention in recent years, mainly due to the fast development of pre-trained language models such as BERT ... | arxiv_0185188 |
Enhanced longevity of the spin helix in low-symmetry quantum wells
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In a semiconductor, collective excitations of spin textures usually decay rather fast due to D’yakonov-Perel’ spin relaxation. The latter arises from spin-orbit coupling, which induces w... | arxiv_0185194 |
Transversals via regularity
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Given graphs *G*1, …, *G**s* all on the same vertex set and a graph *H* with *e*(*H*) ≤ *s*, a copy of *H* is *transversal* or *rainbow* if it contains at most one edge from each *G**c*. We study the case when *H* is spanning and explore how the regularity blow-up... | arxiv_0185197 |
Assoc.* **85** 398–409.
Gelman, A. and Shirley, K. (2010). Inference from simulations and monitoring convergence. In *Handbook of Markov Chain Monte Carlo: Methods and Applications*. (S. Brooks, A. Gelman, G. Jones and X.-L. Meng, eds.) 131–143. Chapman and Hall/CRC Press, Boca Raton, FL.
Ghorpade, S. R. and Limaye, B... | arxiv_0185199 |
Oxygen-bearing organic molecules in comet 67P’s dusty coma: first evidence for abundant heterocycles
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### Received XXX / Accepted YYY
Introduction
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Along with meteorites and Interplanetary Dust Particles, co... | arxiv_0185202 |
Evaluation Measures for Quantification: An Axiomatic Approach
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### Received: / Accepted: DD MM YYYY
Quantification is the task of estimating, given a set *σ* of unlabelled items and a set of classes C = {*c*1, …, *c*∣C∣}, the prevalence (or “relative frequenc... | arxiv_0185203 |
1). The critical surface mass density is, therefore, a characteristic value for the surface mass density which is the dividing line between “weak” and “strong” lenses.
Gravitational lensing conserves surface brightness, but the shape and size of the images differs from those of the source. The distortion of the images ... | arxiv_0185210 |
we present the relationship between specific angular momentum and stellar mass along the evolution of ETG progenitors. Note that actually the angular momentum has been rescaled by the factor *E*1/6(*z*), to remove the trivial redshift evolution associated to the halo angular momentum $j\_{\rm H}$, cf. Eq. ([eq|jhalo])... | arxiv_0185214 |
p* enjoys a payoff of
Φ\*(*p*) = max{Φ(*p*), *U**r*(*p*)},
with the DM indifferent between experimentation and action *r* at the right stopping boundary *p**r**B*. We will call Φ\*(*p*) the *Bayesian value function* throughout. Its characterization is relegated to.
illustrates graphically the value of the Bayesia... | arxiv_0185216 |
per intersection. For a better comparison, we show the average delay (around 40 s) resulting from the fixed-time control in the field with a dashed line when the demand is set to the field demand. As we can see, for both demands, the mean-BP (BP using a prediction method of “mean”) has the highest delay. With a higher... | arxiv_0185221 |
Existence of minimizers and convergence of critical points for a new Landau-de Gennes energy functional in nematic liquid crystals
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The Landau-de Gennes energy in nematic liquid crystals dep... | arxiv_0185222 |
No-go theorem for topological insulators and sure-fire recipe for Chern insulators
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For any symmorphic magnetic space group *G*, it is proven that topological band insulators with vanishing first Chern numbers cannot have a groundstate co... | arxiv_0185223 |
Hyperbolic cross approximation in infinite dimensions
=====================================================
We give tight upper and lower bounds of the cardinality of the index sets of certain hyperbolic crosses which reflect mixed Sobolev-Korobov-type smoothness and mixed Sobolev-analytic-type smoothness in the infini... | arxiv_0185225 |
2(\check g(x,\cdot)) = \cO(h^{-3})$ and $\AIV = \cO(s n^{-(s+1)/s} h^{-3})$.
The function $\check g(x,\cdot)$ is continuously differentiable, Lipschitz and monotone on [0, 1] and so applies to it, yielding $V\_2^2(\check g(x,\cdot)) = \cO(h^{-3})$. The rest follows from.
This corollary tells us that that when the dime... | arxiv_0185227 |
quad W\_{i+130}={\rm{T}}^i W\_{130},\quad
i=1,\ldots 5 \,,\nonumber \\
W\_{136}=\eps\_{1245},\quad W\_{i+136}={\rm{T}}^i W\_{136},\quad
i=1,\ldots 2 \,.\end{gathered}$$
Note that *W*136 is invariant under $\rm T^3$. Under the parity transformation, a pesudo scalar transforms as *ε**i**j**k**l* → − *ε**i**j**k**l* bu... | arxiv_0185228 |
pristine’ profile for ISM-like small amorphous silicate grains (sub-*μ*m). NGC 1333 shows a long wavelength wing similar to that of the other regions which suggests similar grain sizes in the optically thin dust.
Discussion of PTDs and TDs
==========================
We have examined two ways of identifying TDs and PTDs... | arxiv_0185230 |
==================================================================================
Results in the previous two sections might suggest conjecturing that $\Hom\_\bullet^\Cube(G)$ and $\Hom\_\bullet^\Path(G)$ are the same for all graphs *G*. In this section we construct an example showing that this is not always the case.... | arxiv_0185235 |
Hamiltonian reduction of Vlasov-Maxwell to a dark slow manifold
===============================================================
We show that nonrelativsitic scaling of the collisionless Vlasov-Maxwell system implies the existence of a formal invariant slow manifold in the infinite-dimensional Vlasov-Maxwell phase space... | arxiv_0185236 |
-rotated component. The red line is the result using the parameters estimated from fitting to the 2D shear field. The vertical dashed green line marks the projected distance between the two clusters in the low-redshift lens plane.
The centering probabilities $P\_{\rm cen}$ for clusters in the pair 127 + 9596 (upper pan... | arxiv_0185242 |
**Universidade Estadual de Campinas**
Instituto de Física “Gleb Wataghin”
Carolina Arruda Moreira
**Synchronization induced by external forces in modular networks**
**Sincronização induzida por forças externas em redes modulares**
**CAMPINAS
2020**
**Carolina Arruda Moreira**
**Synchronization ... | arxiv_0185246 |
at (1.70878,0.964992); (82) at (1.51863,1.07187); (83) at (1.19925,2.11142); (84) at (1.1396,2.08719); (85) at (1.13587,2.09651); (86) at (1.37075,0.231772); (87) at (1.27879,0.272783); (88) at (0.744405,1.45774); (89) at (0.731977,1.46085); (90) at (0.722656,1.40803); (91) at (0.774852,1.45153); (92) at (1.35149,0.74... | arxiv_0185254 |
dex (or more, for specific models). This allowed us to obtain a rough estimate on the associated uncertainty, as displayed in Table [tabspectra] and Figs. [HD36512] to [CYGOB2-7].
Our sub-sample comprises three dwarf and three supergiant O stars, observed with different instruments. However, all spectra cover the wave... | arxiv_0185258 |
Self-learning Machines based on Hamiltonian Echo Backpropagation
================================================================
A physical self-learning machine can be defined as a nonlinear dynamical system that can be trained on data (similar to artificial neural networks), but where the update of the internal degr... | arxiv_0185259 |
.125 & 0.401
& E & -0.012 & 0.173 & -0.002 & 0.180 & -0.126 & 0.400
& F & -0.006 & 0.124 & -0.009 & 0.098 & -0.119 & 0.405
& G & -0.010 & 0.128 & -0.008 & 0.102 & -0.121 & 0.404
Men & Null & -0.145 & 0.498 & -0.170 & 0.330 & -1.019 & 0.435
& A & -0.106 & 0.279 & -0.177 & 0.377 & -1.016 & 0.434
& B & -0.055 ... | arxiv_0185261 |
}$$
where *ε* = *M**E**l**k**o*2/*k*2. It is obvious that the asymptotic behavior is the same with ([KKmodeinRSmode]) and the general solution is
$$\begin{aligned}
\alpha\_{n}(z)&=&C\_1H(z)M\_{\frac{1}{2},-\sqrt{5+2\epsilon}}\left(\text{i}\frac{2m\_{n}}{k}(k|z|+1)\right)\nonumber\\
&+&C\_2H(-z)M\_{\frac{1}{2},\sq... | arxiv_0185278 |
Orthogonal Policy Learning Under Ambiguity[1](#fn1)
===================================================
Job Market Paper
[[ Click here for the latest version]](https://rdadamo.github.io/mywebsite/JMP_Dadamo.pdf)
This paper studies the problem of estimating individualized treatment rules when treatment effects are par... | arxiv_0185288 |
=1
IPPP/11/62
DCPT/11/124
**Natural Vacuum Alignment from Group Theory: The Minimal Case**
Martin Holthausen[1](#fn1)(*a*) and Michael A. Schmidt[2](#fn2)(*b*)(*c*)
*(*a*) Max-Planck Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany*
*(*b*) Institute for Particle Physics Phenomenology (I... | arxiv_0185294 |
Nonlinear static isogeometric analysis of arbitrarily curved Kirchhoff-Love shells
==================================================================================
Abstract
========
The geometrically rigorous nonlinear analysis of elastic shells is considered in the context of finite, but small, strain theory. The re... | arxiv_0185296 |
7006660917) (40.0,0.9569537905431716) (41.0,0.94852848471399) (42.0,0.938588652087455) (43.0,0.9269173570719071) (44.0,0.9132886332698913) (45.0,0.8974763545798369) (46.0,0.8792666600235295) (47.0,0.8584740576134586) (48.0,0.8349607667482306) (49.0,0.8086580589887477) (50.0,0.7795874111061434) (51.0,0.7478783958711802)... | arxiv_0185299 |
Bounding Helly numbers via Betti numbers[1](#fn1) [2](#fn2) [3](#fn3)
=====================================================================
8cm
*Dedicated to the memory of Jiří Matoušek, wonderful teacher, mentor, collaborator, and friend.*
We show that very weak topological assumptions are enough to ensure the existe... | arxiv_0185303 |
: = *I*(Φ*γ**ɛ*− 3/2*t**ɛ*(*z*)), as indicated in the footnote for Theorem [th:diffusion-process-all-strip], then in the reminder of the proof we have: in *A**u**ɛ* and *A**d**ɛ* we have *γ**ɛ*− 1/2 instead of *ɛ*− *γ* + 1; in *A**u**u**ɛ*, *A**u**d**ɛ*, *A**d**u**ɛ*, *A**d**d**ɛ* we have 2*γ**ɛ*− 1/2 instead of 2*ɛ*−... | arxiv_0185307 |
*/*N*, the distribution of the eigenvalues *λ**n* tends to the Marčenko-Pastur distribution *ρ*(*λ*)*d**λ*. Therefore, for large values of *λ*0 ∼ *N* ∼ *T*, the correction to *λ*0 can be estimated as
$$\lambda\_0' \simeq \lambda\_0+\frac{ \lambda\_0\sigma^2}{T/N}\int\!d\lambda\frac{\rho(\lambda)}{\lambda\_0-\lambda}... | arxiv_0185308 |
V994 Her: A Unique Triply Eclipsing Sextuple Star System
========================================================
### Accepted XXX. Received YYY; in original form ZZZ
[firstpage]
We report the discovery with *TESS* of a third set of eclipses from V994 Herculis (TIC 424508303), previously only known as a doubly-eclipsin... | arxiv_0185320 |
*h* − 1) in $L\_{\bs^D}(\blam^D)$, we clearly get two dual abaci. Since $\mathcal F$ is a finite set, by repeating this process finite many times, abacus $L\_{\bs}(\blam)$ arrives to its core.
Furthermore, as a direct corollary of the above, we can obtain the result on move vectors and we complete the proof.
Frame of t... | arxiv_0185324 |
lowest luminosity powering star (VLA4 which is B2) corresponds to a ZAMS mass of ∼ 9M⊙, found using the ZAMS luminosities of and the theoretical HR diagram of. Thus if this cluster has a fully populated IMF, extrapolating a Krupa IMF down to 1M⊙ suggests there should be ∼ 100 stars in this cluster with mass above ... | arxiv_0185325 |
C^1} & {\color{green!70!black}C^{12}}\arrow{l}[swap]{l^2}\\
{\color{red}C^\emptyset} \arrow{u}{r^1} & {\color{orange}C^2} \arrow{l}{l^2}\arrow{u}[swap]{r^1}
\end{tikzcd}$$
and $\chi\_2\colon\hofib(r^2\_\*)\to\Omega\hofib(l^2\_\*)$ is depicted in, with the colours indicating the ambient spaces from. Note that *f*1 is... | arxiv_0185326 |
Myhill-Nerode methods for hypergraphs[1](#fn1)
==============================================
#### Abstract.
We give an analog of the Myhill-Nerode methods from formal language theory for hypergraphs and use it to derive the following results for two NP-hard hypergraph problems.
* We provide an algorithm for testing wh... | arxiv_0185339 |
2. K. I. O. acknowledges funding from the Simons Collaboration on the Origins of Life (SCOL) and the David and Lucile Packard Foundation. Astrochemistry in Leiden is supported by the EU A-ERC grant 291141 CHEMPLAN.
natexlab#1#1
Adams, N. G., & Smith, D. 1985,, 294, L63
Aikawa, Y., Furuya, K., Nomura, H., & Qi, C. 2015,... | arxiv_0185344 |
can be decoded, treating the desired signal as noise, and subtracted from the received signal, before the desired signal is decoded. We propose efficient methods to compute the Pareto-optimal rate points and corresponding beamforming vector pairs, by maximizing the rate of one link given the rate of the other link. We... | arxiv_0185346 |
The Hot Interstellar Medium in Normal Elliptical Galaxies. II.Morphological Evidence for AGN Feedback
=====================================================================================================
We report on the discovery of a new quantitative relationship between X-ray gas morphology and radio and X-ray AGN l... | arxiv_0185351 |
– (-0.4,-0.4); (-2.6,-0.4) circle (6.0pt); (-2.5,-0.4) circle (6.0pt); (-2.4,-0.4) circle (6.0pt); (-2.3,-0.4) circle (6.0pt); (-2.2,-0.4) circle (6.0pt); (-2.1,-0.4) circle (6.0pt); (-2.,-0.4) circle (6.0pt); (-1.9,-0.4) circle (6.0pt); (-1.8,-0.4) circle (6.0pt); (-1.7,-0.4) circle (6.0pt); (-1.6,-0.4) circle (6.0pt... | arxiv_0185353 |
* emitters. We study 59 luminous H*α* emitters with $L\_{\rm H\alpha}>$$L\_{\rm H\alpha}^\*$, roughly equally split per redshift slice at *z* ∼ 0.8, 1.47 and 2.23 from the HiZELS and CF-HiZELS surveys. We find that, overall, 30 ± 8% are AGN (80 ± 30% of these AGN are broad-line AGN, BL-AGN), and we find little to no ev... | arxiv_0185357 |
Quantum dot spin qubits in Silicon: Multivalley physics
=======================================================
Research on Si quantum dot spin qubits is motivated by the long spin coherence times measured in Si, yet the orbital spectrum of Si dots is increased as a result of the valley degree of freedom. The valley de... | arxiv_0185362 |
1,0) rectangle (2,1); (1,3) rectangle (2,4); (1,5) rectangle (2,6); (2,0) rectangle (3,1); (2,3) rectangle (3,4); (2,5) rectangle (3,6); (3,0) rectangle (4,1); (3,4) rectangle (4,5); (3,5) rectangle (4,6); (4,0) rectangle (5,1); (4,1) rectangle (5,2); (4,2) rectangle (5,3); (4,4) rectangle (5,5); (4,5) rectangle (5,6);... | arxiv_0185363 |
Fast-Rolling Relaxion
=====================
We discuss new mechanisms to stop the relaxion field during inflation. They can be realized in a generic model, including the original model but in a quite different parameter region. We consider a *fast-rolling* relaxion field, which can go over the bumps created by QCD-like... | arxiv_0185365 |
Disc-binary interactions in depleted post-AGB binaries
======================================================
### Received 4 May 2020 / Accepted 16 August 2020
Introduction
============
The evolution of evolved stars in binary systems is still a poorly understood topic in astronomy. Consequently, understanding the diff... | arxiv_0185371 |
^{\theta'}}\pi\_{0}^{\theta'}(\omega')\ell\_t(\theta',\omega')},\label{eq:likelihood-ratio}\end{aligned}$$
where
$$\begin{aligned}
\ell\_t(\theta,\omega)\coloneqq \prod\_{\tau=0}^{t-1}q^{\theta}\left(y\_{\tau}|a\_{\tau},\omega\right),\label{eq:likelihood}\end{aligned}$$
and ℓ*t*(*θ*ʹ, *ω*ʹ) is defined analogousl... | arxiv_0185372 |
correction term, proportional to the Gauss-Bonnet coupling parameter. The correction term modifies the spacetime structure, in particular, the location of the event horizon. Proceeding further, we have derived effective field equations for *f*(*R*) gravity with Gauss-Bonnet correction term and a static spherically sym... | arxiv_0185373 |
to Navier boundary conditions with nonzero friction coefficient, such as were treated in.
Acknowledgments
---------------
The authors would like to thank an anonymous referee for for the suggestion to use conormal Sobolev spaces. A.V.B. and D.I. are grateful for the hospitality of the Universidade Federal do Rio de J... | arxiv_0185379 |
and orbital motion. The Roche lobe surface potentials are determined by adjusting them until the radii of both components correspond to the literature values presented in Table [tab:StellarParam]. While this does allow for the stellar shape to deviate from spherical, we find no significant deviation from spherical geo... | arxiv_0185381 |
IRAS 09002-4732: A Laboratory for the Formation of Rich Stellar Clusters
========================================================================
IRAS 09002-4732 is a poorly studied embedded cluster of stars in the Vela Molecular Ridge at a distance of 1.7 kpc. Deep observations with the *Chandra* X-ray Observatory, co... | arxiv_0185382 |
Qubit-excitation-based adaptive variational quantum eigensolver
===============================================================
Abstract
========
Molecular simulations with the variational quantum eigensolver (VQE) are a promising application for emerging noisy intermediate-scale quantum computers. Constructing accurat... | arxiv_0185387 |
array}} \right] + \left[ {\begin{array}{\*{20}{c}}
{{c\_1} + {c\_2}}&{ - {c\_2}}\\
{ - {c\_2}}&{{c\_2}}
\end{array}} \right]\left[ {\begin{array}{\*{20}{c}}
{{{\dot X}\_1}}\\
{{{\dot X}\_2}}
\end{array}} \right] + \left[ {\begin{array}{\*{20}{c}}
{{k\_1} + {k\_2}}&{ - {k\_2}}\\
{ - {k\_2}}&{{k\_2}}
\end{array}... | arxiv_0185402 |
Taking measurements of the kinematic Sunyaev-Zel’dovich effect *forward*: including uncertainties from velocity reconstruction with forward modeling
====================================================================================================================================================
Introduction
=========... | arxiv_0185404 |
Does the Markov Decision Process Fit the Data: Testing for the Markov Property in Sequential Decision Making
============================================================================================================
The Markov assumption (MA) is fundamental to the empirical validity of reinforcement learning. In this... | arxiv_0185408 |
the same window.
### CSR in gas above cutoff value
The simplest null hypothesis is that there is no correlation between molecular cloud material and YSOs. CSR is unlikely to be a successful model as (i) protostars and prestellar cores are known to be associated with dense material within molecular clouds, (ii) the mea... | arxiv_0185410 |
Viscosity of a sheared correlated (near-critical) model fluid in confinement
=============================================================================
Second-order phase transitions are characterized by a divergence of the spatial correlation length of the order parameter fluctuations. For confined systems, this i... | arxiv_0185413 |
1.0truecm
D. R. Grigore, [1](#fn1)
Department of Theoretical Physics, Institute for Physics and Nuclear Engineering ``Horia Hulubei"
Institute of Atomic Physics
Bucharest-Măgurele, P. O. Box MG 6, ROMÂNIA
Quantum theory of the gravitation in the causal approach is studied up to the second order of perturbation theory. ... | arxiv_0185420 |
Combined analysis of neutron star natal kicks using proper motions and parallax measurements for radio pulsars and Be X-ray binaries
====================================================================================================================================
### Accepted XXX. Received YYY; in original form ZZZ
[... | arxiv_0185424 |
On 2-form gauge models of topological phases
============================================
Introduction
============
Over the last several decades, *quantum field theories* have emerged as the central language in which modern theoretical physics is formulated. For instance, *quantum phases of matter* may succinctly be d... | arxiv_0185429 |
λ*/2 = 15 GHz. The finite-element solution was carried out with as fine a mesh as reasonably possible on a PC with 8 GB of memory; this typically resulted in more than 300,000 first-order elements, and small (less than 0.05%) variations in computed *S*-parameters as the mesh was refined to its final size, so high accur... | arxiv_0185430 |
How does the mass and activity history of the host star affect the population of low-mass planets?
==================================================================================================
### Accepted XXX. Received YYY; in original form ZZZ
[firstpage]
The evolution of the atmospheres of low and intermediate-... | arxiv_0185431 |
.4847. Avellaneda, M., & Stoikov, S. (2008). High-frequency trading in a limit order book. Quantitative Finance, 8(3), 217-224. Balseiro, S., & Candogan, O. (2015). Optimal Contracts for Intermediaries in Online Advertising. Available at SSRN 2546609. Balseiro, S. R., Feldman, J., Mirrokni, V., & Muthukrishnan, S. (201... | arxiv_0185432 |
Toward a first-principle derivation of confinement and chiral-symmetry-breaking crossover transitions in QCD
============================================================================================================
We give a theoretical framework to obtain a low-energy effective theory of quantum chromodynamics (QCD... | arxiv_0185433 |
U* such that the rank of $U + \overline{\pi\Lambda}\_R$ is 6, which is equivalent to $U = \overline{\pi\Lambda}\_R$. Observe that *π*Λ ⊗ *W*(F) is a *τ*-stable, self-dual lattice, hence it belongs to NL(F) for some vertex lattice of type 6 contained in Λ, and it corresponds to a *p*-divisible group *X**π*Λ ∈ NΛ(F).
The... | arxiv_0185437 |
Non-uniqueness for the transport equation with Sobolev vector fields
====================================================================
We construct a large class of examples of non-uniqueness for the linear transport equation and the transport-diffusion equation with divergence-free vector fields in Sobolev spaces *... | arxiv_0185438 |
425832748) (2,0.381754369) (3,0.32449168) (4,0.270150423) (5,0.236228347) (6,0.236945167) (7,0.279601574) (8,0.362974435) (9,0.478059709) (10,0.610718846) (11,0.745231509) (12,0.867584109) (13,0.967576861) (14,1.039378405) (15,1.08077383) (16,1.091810465) (17,1.073650479) (18,1.028208375) (19,0.958657861) (20,0.8703966... | arxiv_0185443 |
The Category of Factorization
=============================
We introduce and investigate the category of factorization of a multiplicative, commutative, cancellative, pre-ordered monoid *A*, which we denote F(*A*). The objects of F(*A*) are factorizations of elements of *A*, and the morphisms in F(*A*) encode combinato... | arxiv_0185444 |
Chiral strings, the sectorized description and their integrated vertex operators
================================================================================
A chiral string can be seen as an ordinary string in a singular gauge for the worldsheet metric and has the ambitwistor string as its tensionless limit. As pr... | arxiv_0185447 |
A link condition for simplicial complexes, and CUB spaces
=========================================================
0.8 Abstract. We motivate the study of metric spaces with a unique convex geodesic bicombing, which we call CUB spaces. These encompass many classical notions of nonpositive curvature, such as CAT(0) spac... | arxiv_0185450 |
Distributed Stochastic Subgradient Projection Algorithms for Convex Optimization
================================================================================
We consider a distributed multi-agent network system where the goal is to minimize a sum of convex objective functions of the agents subject to a common conve... | arxiv_0185453 |
The mass function of young star clusters in spiral galaxies
===========================================================
The initial cluster mass function (ICMF) in spiral discs is constrained and compared with data for old globular clusters and young clusters in starbursts. For a given absolute magnitude, the cluste... | arxiv_0185463 |
tighter confidence bounds on the J*i*, *f**i*, *g**i*, 1, and *g**i*, 2 than the models with only one constraint per multiple image. Hence, to obtain the tighest confidence bounds on local lens properties, Grale requires as many constraints as possible in the vicinity of the point where the lens properties are to be d... | arxiv_0185464 |
Composite 2HDM with singlets: a viable dark matter scenario
===========================================================
Introduction
============
The spectacular success of the Standard Model (SM) in describing particle physics phenomena, culminated with the discovery of the Higgs boson, still leaves us with several op... | arxiv_0185466 |
Expressivity Landscape for Logics with Probabilistic Interventionist Counterfactuals
====================================================================================
Causal multiteam semantics is a framework where probabilistic dependencies arising from data and causation between variables can be together formalize... | arxiv_0185472 |
Catalog II (Bucciarelli et al. 2001). Photographic photometry from the first-epoch plates calibrated on Tycho2 is adopted when no astrograph CCD photometry is available, (see Girard et al. 2011).
Proper Motions
==============
Proper motions are determined using the central-plate overlap method (e.g., Girard et al. 198... | arxiv_0185473 |
Galaxy-scale ionised winds driven by ultra-fast outflows in two nearby quasars
==============================================================================
### Received ; accepted
Introduction
============
Feedback from active galactic nuclei (AGNs) is expected to have a significant impact on the formation and evolut... | arxiv_0185501 |
the stochastic differential game can be thought as an optimization problem of a central planner, who is looking for a common strategy in order to optimize some collective objective functional.
The state space of mean field control problems is the set of probability measures, and usually the Wasserstein space $\Pc\_2(\... | arxiv_0185502 |
Nearly finite Chacon transformation
===================================
We construct an infinite measure preserving version of Chacon transformation, and prove that it has a property similar to Minimal Self-Joinings in finite measure: its Cartesian powers have as few invariant Radon measures as possible.
**Keywords:** ... | arxiv_0185508 |
Ricci flow on three-dimensional, unimodular metric Lie algebras
===============================================================
=1
We give a global picture of the Ricci flow on the space of three-dimensional, unimodular, nonabelian metric Lie algebras considered up to isometry and scaling. The Ricci flow is viewed as ... | arxiv_0185511 |
Accretion Flow Evolution of a New Black Hole Candidate MAXI J1348-630 During the 2019 Outburst
==============================================================================================
Galactic black hole candidate MAXI J1348-630 was recently discovered by *MAXI* and *Swift*/BAT satellites during its first outburs... | arxiv_0185513 |
ALBERTI, a Multilingual Domain Specific Language Model for Poetry Analysis
==========================================================================
[orcid=0000-0002-9143-5573, email=versae@nb.no, ]
[orcid=0000-0001-7116-9338, email=alvaro.perez@linhd.uned.es, ]
[orcid=0000-0001-6330-4958, email=sros@scc.uned.es, ]
[ ... | arxiv_0185515 |
to a conversion factor that is very close to the average of the individual *I**C**O*-to-$N(\h2)$ ratios: *X*(CO) = $(1.9\pm0.2)\times10^{19}\,\cm2/\kkms$. The average *X*-factor applicable to galaxy centers is quite robust and well defined even when individual CO-to-hydrogen conversion factors are still subject to unc... | arxiv_0185517 |
The linear hidden subset problem for the (1+1) EA with scheduled and adaptive mutation rates
============================================================================================
We study unbiased (1 + 1) evolutionary algorithms on linear functions with an unknown number *n* of bits with non-zero weight. Static ... | arxiv_0185518 |
the condition that **x** be continuously differentiable implies the vector relations
$$\begin{aligned}
\mathbf{x}\_1(\varphi) &= \mathbf{x}\_2(\varphi), \label{eq:gerver-cont12} \\
\mathbf{x}\_1'(\varphi) &= \mathbf{x}\_2'(\varphi), \label{eq:gerver-diff12} \\
\mathbf{x}\_2(\theta) &= \mathbf{x}\_3(\theta), \label{... | arxiv_0185520 |
X-Rays Beware: The Deepest Chandra Catalogue of Point Sources in M31
====================================================================
### Received; Accepted
[firstpage]
This study represents the most sensitive *Chandra* X-ray point source catalogue of M31. Using 133 publicly available *Chandra* ACIS-I/S observation... | arxiv_0185523 |
Symmetries of Fano varieties
============================
We study Fano varieties endowed with a faithful action of a symmetric group, as well as analogous results for Calabi–Yau varieties, and log terminal singularities. We show the existence of a constant *m*(*n*), so that every symmetric group *S**k* acting on an *n... | arxiv_0185530 |
Total and linearly polarized synchrotron emission from overpressured magnetized relativistic jets
=================================================================================================
We present relativistic magnetohydrodynamic (RMHD) simulations of stationary overpressured magnetized relativistic jets whic... | arxiv_0185540 |
the number density of obstacles plays only a subordinate role. Since an obstacle can, in principle, trap arbitrarily many active particles, it is irrelevant whether a single obstacle extracts power from all active particles together or whether several obstacles each extract only a fraction of the power. The only requi... | arxiv_0185542 |
A Theory of Unsupervised TranslationMotivated by Understanding Animal Communication
===================================================================================
Neural networks are capable of translating between languages—in some cases even between two languages where there is little or no access to parallel tra... | arxiv_0185545 |
Generalized Inverse Based Decoding
==================================
The concept of Generalized Inverse based Decoding (GID) is introduced, as an algebraic framework for the syndrome decoding problem (SDP) and low weight codeword problem (LWP). The framework has ground on two characterizations by generalized inverses ... | arxiv_0185546 |
Detection of radio-AGN in dust-obscured galaxies using deep uGMRT radio continuum observations
==============================================================================================
[
Radio observations being insensitive to the dust-obscuration, have been exploited to unveil the population of Active Galactic Nu... | arxiv_0185547 |
where ${\rm Int}[x]$ denotes nearest integer to *x*), as a function of frequency. The corresponding plot is shown in the left panel of Fig. [longa]. Here, instead of looking at equal number cycles *n*0, we study the behavior of the quantities averaged over equal span of stroboscopic time *n*0*T*. This is done in the re... | arxiv_0185554 |
Parity-violating scalar-tensor theory and the Qi-Xiu
====================================================
Introduction
============
While the dark sector of current cosmology remains insufficiently understood, new physics beyond general relativity (GR) and the standard model of particle physics, including the strong, e... | arxiv_0185560 |
varepsilon}\_h}]\_E/\partial s \, \nu\_E)$ and its extension *P**E**ρ**E* define the test function *ϕ* : = *b**P**E**ρ**E* ∈ *C*0∞(*ω**E*) in Lemma [lemma:representation]. The representation formula and (*ɛ*(*v*), Curl2*ϕ*)*L*2(*ω**E*) = 0 lead to
$$\| b^{1/2} \rho\_E\|\_{L^2(E)}^2
= (\varepsilon(v)-{\bm{\varepsilo... | arxiv_0185565 |
Two-way classical communication remarkably improves local distinguishability
============================================================================
We analyze the difference in the local distinguishability among the following three restrictions; (i) Local operations and only one-way classical communications (one-... | arxiv_0185567 |
3^{-1} = \frac{1}{4}, \\
\frac{{W^{\*,}}'\_{t\_1}(0, {\phi}\_{0} )}{\gamma(W\_{{\mathfrak{p}}\_1})} &= \frac{2o(t\_1) + 1}{2} \log(3) \text{ when } 2\mid o(t\_1)\end{aligned}$$
when *μ* ∈ *S*1 as $\alpha(\mu, t\_2) = -\frac{{\mathrm{d}}\_2 {{\mathrm{Nm}}}(\mu)}{3\delta} - t\_2 \in \frac{{\mathrm{d}}\_2}{3} - t\_2 +... | arxiv_0185574 |
5) and (-0.2,1.8).. (0,1.8).. controls (0.2,1.8) and (0.3,1).. (0.3,1.2) ; (-2.3,1.1).. controls (-2.3,2) and (-1.55,2).. (-1.45,1.75).. controls (-1.35,1.25) and (-1.7,1.5).. (-1.7,1).. controls (-1.7,-0.2) and (0,-0.8).. (0.5,-0.8).. controls (2,-0.8) and (1.8,0).. (1.3,1.3).. controls (1.3,1.8).. (2,1.8).. controls ... | arxiv_0185578 |
5]; at (-0.25-3.5,-0.75)[circle,draw=black,fill=black,scale=0.5]; at (-0.75-3.5,-0.75)[circle,draw=black,fill=black,scale=0.5]; at (0.25-3.5,1.25)[circle,draw=black,fill=black,scale=0.5]; at (-0.25-3.5,1.25)[circle,draw=black,fill=black,scale=0.5]; at (0.25-3.5,-1.25)[circle,draw=black,fill=black,scale=0.5]; at (-0.25-... | arxiv_0185581 |
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